- Email: [email protected]
Controllable postures of a dual-crawler-driven robot
Mechatronics 20 (2010) 281–292
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Controllable postures of a dual-crawler-driven robot Qiquan Quan, Shugen Ma * Department of Robotics, Ritsumeikan University, Japan
a r t i c l e
i n f o
Article history: Received 5 August 2009 Accepted 5 January 2010
Keywords: Tracked robot Crawler mechanism Quasi-static analysis Realizable postures Posture transition
a b s t r a c t This paper presents a tracked robot equipped with the proposed crawler mechanism, in which a planetary gear reducer is employed as a transmission device and provides two outputs in different forms with only one actuator. A robot which contains two crawler modules can generate several configurations through cooperatively controlling the two actuators. This tracked robot, which uses two actuators to give four outputs, however could have less realizable postures than that where each output is provided by one actuator exactly. To figure out what postures can be generated by the introduced dual-crawler robot, quasi-static analysis of the robot has been conducted while taking the rolling resistance into consideration and its realizable postures have been obtained numerically. The posture transition of the robot is also discussed in this paper. Experiments are conducted to verify the quasi-static analysis for each configuration. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction Development of an efficient mobile mechanism for robots that are required to operate in irregular environments is always an important task. Robots using traditional wheeled mobile mechanisms can be programmed to travel over relatively smooth terrain easily; however, mobility over rugged terrain is limited by the diameter of wheels of robot [1,2]. Robots using legged mobile mechanisms can move well on uneven terrain, but they encounter several challenges, including difficulty of control and lack of stability [3,4]. Since tracked mobile mechanisms have advantages, like excellent stability, low pressure to terrain and simplicity of control, they have been widely deployed in irregular environments. Tracked mobile mechanisms, nevertheless, are still somewhat limited due to some mechanism parameters, such as the diameter of the front sprocket [5,6]. The most common way to improve the mobility and adaptability of tracked mobile mechanisms is to build a multi-track robot by linking several active or passive units in serial or parallel way [7–9]. However, to provide assisting actions and control the system correctly, it is necessary to add some extra actuators, mechanisms and control elements. In other words, these assisting actions cannot be performed autonomously. To resolve the difficulties outlined above, we have proposed a crawler mechanism with polymorphic locomotion [10,11]. This mechanism, which is equipped with a planetary gear reducer, makes use of only one actuator to provide two outputs. By determining the reduction ratio of two outputs in a suitable proportion, * Corresponding author. E-mail addresses: [email protected] (Q. Quan), [email protected]. ac.jp (S. Ma).
the crawler is capable of switching autonomously between two locomotion modes according to the terrain. The main characteristic of the mechanism is that the polymorphic locomotion is provided by one actuator and switching between the two modes of locomotion occurs autonomously. In this paper, we will introduce a tracked robot comprised of two modules of this crawler mechanism. The introduced tracked robot can generate several configurations by controlling properly the two actuators as well as each module’s polymorphic motions. However, this tracked robot, which employs two actuators to provide four outputs, would have less realizable postures than that where each output is produced by an individual actuator. This robot still has an advantage in decreasing the impact to actuators. When a robot moves over rough terrain, it inevitably collides with various obstacles, creating an impact effect on the driving actuator. When there is a collision transmitted from one output, another output can release part of the impact energy in our crawler mechanism. This impact absorption of the mechanism makes the actuator subject to less impact, and thus more safe [12]. The posture analysis of the dual-crawler-driven robot has been conducted without considering rolling resistance and did not discuss the posture transition [13]. This paper will give a systematical explanation of the controllable postures in both cases: ‘‘without moving velocity” and ‘‘with moving velocity”, and discuss the related posture transition. This paper presents static analysis of the robot to find out all the realizable postures and their possible transitions. The paper is organized as follows. The proposed crawler mechanism with polymorphic motion is introduced in Section 2 and the dual-crawler-driven robot is also given in this section. Section 3 gives all the possible geometric postures of this robot and the relevant geometric relations. Section 4 conducts the static analysis of the
0957-4158/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2010.01.001
282
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
dual-crawler-driven robot and discusses its realizable postures and posture transitions. Section 5 concludes the paper. 2. Mechanism of a dual-crawler robot By using two modules of the proposed crawler mechanism, a tracked robot with polymorphic locomotion was designed. In this section, we will first briefly describe the proposed crawler mechanism and then introduce the tracked robot consisting of two modules of this crawler mechanism. 2.1. Mechanism of a crawler module The proposed crawler mechanism is capable of providing two kinds of output with just one actuator. The first output is transmitted to the crawler-belt and drives the crawler to move forwards; the second one is employed to drive the connecting frame that links two sprockets of the crawler, as shown in Fig. 1. The planetary gear reducer was adopted as the main power transmission for our crawler mechanism, as shown in Fig. 1a. The input torque of the actuator is transmitted to the sun gear of the planetary gear reducer through a pair of bevel gears. Since the carrier of the planetary gear reducer is linked with an active pulley, the torque is derived from the sun gear and transmitted to the active pulley, the crawler belt, and acts as the first output to drive the crawler mechanism to forward or backward on even ground or slopes. As the second output, the torque is derived from the sun gear and transmitted to the ring gear of the planetary gear reducer, and then to a triangular gear reducer, and then to the connecting frame. The triangular reducer consists of three pairs of spur gears. The rotation of the connecting frame drives the crawler unit to rotate wholly around the input axle. 2.2. Polymorphic locomotion modes To describe the whole locomotion process while the crawler moves in irregular environment, we present three locomotion modes, referred to as ‘‘motion mode”, ‘‘rotation mode” and ‘‘recovering mode”.
(1) Motion mode (1, 2, 6 in Fig. 2): The crawler mechanism moves on an even terrain or slope like a normal tracked vehicle since the power of the actuator is transmitted to crawler-belt. (2) Rotation mode (3, 4 in Fig. 2): When the crawler mechanism contacts an obstacle, since the rotation of the crawler belt is stopped by the resistance from the ground and the power has to be transmitted to the connecting frame, the rotation of connecting frame drives the crawler mechanism to climb over the obstacle. (3) Recovering mode (5 in Fig. 2): Once the crawler mechanism has climbed over the obstacle, the power is transmitted to the connecting frame and drives the crawler mechanism to return back continuously until it recovers to the initial position. To achieve the proposed locomotion autonomously in irregular environments, the power transmission of the crawler should be designed to meet the following three conditions: (1) One motor input gives two outputs in the transmission. (2) The two outputs must rotate in the same direction. (3) The two reducer ratios are selected in a certain range. The mechanism of our crawler model equipped with a planetary gear reducer meets condition (1) and (2). Concerning the most important condition (2), we can determine the proportion of reduction ratios of two outputs within a certain range. The reduction ratios on outputs 1 and 2 have been designed to be 4 and 30, and the two outputs have the same rotation direction [10,14]. In motion mode, to drive the crawler mechanism to move on even ground or slope like a normal tracked vehicle, propulsion on the crawler belt has been designed larger than motion resistance. At the same time, rotation torque on the connecting frame is smaller than the rotation resistance generated by gravity of crawler and payload. In the same way, when the crawler mechanism contacts an obstacle, to climb over the obstacle in the proposed locomotion mode instead of track-slipping, the propulsion on the crawler belt has been designed smaller than the friction resistance so that the
Fig. 1. Mechanism of one crawler module.
283
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Belt
3
2
1
Frame 6
5
4
Fig. 2. Locomotion modes including ‘‘motion mode”, ‘‘rotation mode”, and ‘‘recovering mode” for step climbing.
Table 1 Physical parameters of the robot. Parameter
Fig. 3. Prototype of a dual-crawler-driven robot.
crawler belt can be fixed. Concurrently, the rotation torque is larger than the rotation resistance to lift vehicle body to climb over the obstacle. After the crawler mechanism has climbed up the obstacle, it can recover to the initial position autonomously. 2.3. Prototype of a dual-crawler-driven robot A tracked robot that is realized by connecting two modules of crawler mechanism through the body is shown in Fig. 3. This robot can not only switch autonomously between different locomotion
y
Rear
d3
θr
d4
o4 x
Weight of robot body Weight of frame Weight of active pulley Weight of passive pulley Distance between driven axles of two crawlers Distance between axles of rear active pulley and Center of Gravity (CG) of the body Distance between axles of active and passive pulleys Distance between axle of active pulley and CG of the frame Radius of pulley Radius of sun gear in planetary gear reducer Radius of planetary gear in planetary gear reducer Radius of ring gear in planetary gear reducer Ratio of reducer from motor to pulley Ratio of reducer from motor to triangle gear reducer Ratio of triangle gear reducer Ratio of reducer from motor to frame, i2 ¼ i21 i22 Ratio of reducer ratios ði2 ð30Þ=i1 ð4ÞÞ
β
Unit
G1 G2 G3 G4 d1 d2
1.06 0.14 0.14 0.14 190 95
kg kg kg kg mm mm
d3 d4
65 32.5
mm mm
R r1 r2 r3 i1 i21 i22 i2 K
29 4 4 12 4 3 10 30 7.5
mm mm mm mm
modes adapting to the terrain, but also generate several postures through controlling the two actuators cooperatively. The dimensions of the robot are shown in Fig. 4 and its physical parameters, which will be used in the following analysis are listed in Table 1.
d1
d2
Value
θf
Front
o2
o1
o3 Fig. 4. Dimensions of the dual-crawler-driven robot.
284
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292 Table 2 Incline angle b of robot body for each region of configurations.
θ r = 90°
hr
θ f = 270°
hf [0°, 180°]
[180°, 360°]
[0°, 180°]
b¼0
b ¼ arcsin
[180°, 360°]
hr b ¼ arcsin d3 sin d1
b¼
d3 sin hf d1 d sin hr d3 sin hf arcsin 3 d1
θ f = 90° θ f , θ r = 0°
Fig. 5 shows a sphere to represent the region of all configurations of the robot, where each corresponding configuration is on the surface of the sphere. For easier view, Fig. 6 shows all the configurations in an unfolded plane, where different combinations of the front angle hf and the rear angle hr can generate the relevant geometric possible postures. In total, there are 12 possible typical configurations when both the front angle hf and the rear angle hr vary from 0° to 360°. The different values of incline angle b can be classified into four groups shown in Figs. 5 and 6: Region rð0 6 hf 6 180 ; 0 6 hr 6 180 Þ, Region sð180 6 hf 6 360 ; 0 6 hr 6 180 Þ, Region tð0 6 hf 6 180 ; 180 6 hr 6 360 Þ, and Region uð180 6 hf 6 360 ; 180 6 hr 6 360 Þ. In the coordinate system ðxo4 yÞ of Fig. 4, the coordinates of the center of the rear passive pulley o3 , the center of the front active pulley o2 , the center of the front passive pulley o1 can be expressed as
θ r = 270° Fig. 5. Regions of all the configurations of a dual-crawler-driven robot represented by the latitude and longitude of a sphere.
3. Possible geometric configurations of the robot As shown in Fig. 4, the dual-crawler-driven robot consists of the rear crawler module, the robot body, and the front crawler module. The action that the front module and the rear module keep at different positions, can produce different configurations. Thus, there are several geometric possible postures for this dual-crawler robot. In order to define the configurations of this robot in a twodimensional environment, an orthogonal coordinate system ðxo4 yÞ is established at the center of the rear active pulley, as shown in Fig. 4. The x axis is parallel to the ground surface while the y axis is normal to the ground. The rear angle hr is the angle that the rear crawler module rotates from the x axis to the connecting frame of the rear module; the front angle hf is the angle that the front crawler module rotates from x axis to the connecting frame of the front module. As shown in Fig. 5, hf and hr are selected as the latitude and longitude, respectively, so that the point on the surface of the sphere can represent the corresponding configuration.
xo3 ¼ d3 cos hr ;
yo3 ¼ d3 sin hr
xo2 ¼ d1 cos b;
yo2 ¼ d1 sin b
xo1 ¼ d1 cos b þ d3 cos hf ;
ð1Þ
yo1 ¼ d1 sin b þ d3 sin hf
360
θ r (°)
(1) For the configurations in the Region r, the incline angle of the robot body b is always kept at 0°. (2) For the configurations in the Region s, the rear active pulley maintains full contact with ground surface while the front crawler module is lifted up. Since the centers of o1 and o4 have the same y coordinate element, the incline angle b can thus be described by
12
11 8
4
6
180
5
10
9 3
7
1 0
2 180 Fig. 6. All geometrical configurations of the robot.
360 θ f (°)
285
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
b ¼ arcsinðd3 sin hf =d1 Þ
ð2Þ
(3) For the configurations in the Region t, the front active pulley always stays in contact with ground surface while the rear crawler module is lifted up. The fact that centers of o2 and o3 have the same y coordinate element makes the incline angle b given by
b ¼ arcsinðd3 sin hr =d1 Þ
ð3Þ
(4) For the configurations in the Region u, both the front crawler module and the rear crawler module are lifted up. Because of the fact that centers of o1 and o3 have the same y coordinate element, the incline angle b can be described by
b ¼ arcsinððd3 sin hr d3 sin hf Þ=d1 Þ
ð4Þ
Table 2 summarizes the angle of incline of the robot body b for each different case of configurations. From the above analysis, we can obviously know that the robot can realize several possible configurations. For executing some tasks in rough terrain, the posture of robot body is desired to change according to the task. For example, if a manipulator is mounted on the robot body, as the base of the manipulator, the robot body can help the end-effector to perform the desired tasks. Thus, all the controllable postures should be found in the following analysis. 4. Quasi-static analysis of all configurations of the robot From the geometrical analysis of the dual-crawler-driven robot stated in Section 3, it is known that this robot could generate several configurations through cooperatively controlling the two actuators. However, this tracked robot, which makes use of two
Fo4 py o4
Fb 22
actuators to provide four outputs, could have less realizable postures than that where each output is given by only one individual actuator. In this section, we will conduct the quasi-static analysis of the robot and discuss realizable postures and their transitions. Since there are 12 typical configurations as stated in Section 3, there should be 12 different groups of equations for the statics. In order to get a group of statics equations, which can be employed to describe each posture clearly, we will present a group of basic formulations for the statics analysis.
4.1. Basic statics formulations for all configurations It can be easily found that the difference of each configuration is just that for each module of the robot, the active pulley contacts ground, the passive pulley contacts the ground, or both the active and passive pulleys contact the ground. To describe all the possible configurations in a general form, configuration 12 has been selected to be the basic posture for the general form. We presume that the normal forces and friction are exerted at the rim of each pulley. Thus, this general form can be used to describe all the possible configurations. The difference is that the normal force and frictions at the rim of pulley should be regarded as zero with respect to the different configurations. A slope, which is denoted by a, is also considered in the general form of equations, thereby this general form can describe all the possible configurations in a two-dimensional environment. Still, in the coordinate system, the directions of x axis and y axis are selected to be parallel and perpendicular to the slope, respectively. As shown in Fig. 7, the normal force N r4 and N f 2 , which are normal to the slope, are assumed to be exerted at the rim of the rear active and front active pulleys, respectively. Friction F rc4 and F fc2 , which are parallel to the slope, are also assumed to be exerted at the rim of the rear active and front active pulleys, respectively.
Fo2 py M fp
M rp Fo4 px
Fb11 o2 G3
M4 Frc 4
G3
Ffc 2
Fb 21
Nr 4
Nf2
Fo2 px M2
Fb12
o1
o3 α Fig. 7. Force diagram of the rear and front active pulleys.
Fo4 fy
Fo2 fy Fo4 fx
o4
Fo2 fx
o2 M rf o3
Fo3 x
M ff G2
Fo1 x
o1 Fo1 y
G2 Fo3 y Fig. 8. Force diagram of the rear and front frames.
286
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
where Mrf and Mff are the second outputs of each module to drive the rear and front frame, respectively. F o3 x and F o3 y are the interaction forces between the rear frame and rear passive pulley in x and y direction. F o4 fx and F o4 fy are the interaction forces between the rear frame and robot body in x and y direction. F o1 x and F o1 y are the interaction forces between the front frame and front passive pulley in x and y direction. F o2 fx and F o2 fy are the interaction forces between the front frame and robot body in x and y direction. A force diagram concerning the front and rear passive pulley is shown in Fig. 9. The equilibrium of the two passive pulleys can be obtained by
Rolling resistance M 4 and M 2 are assumed to be exerted likewise at the rim of front and rear active pulleys. As shown in Fig. 7, additional forces including normal force and friction are exerted at the rim of the rear active and front active pulleys, respectively. The equilibrium of the rear active and front active pulleys can be expressed as
8P > < P F x ¼ F o4 px þ ðF b21 þ F b22 Þ cos hr þ F rc4 G3 sin a ¼ 0 F y ¼ F o4 py þ ðF b21 þ F b22 Þ sin hr þ Nr4 G3 cos a ¼ 0 > :P Mo4 ¼ F b21 R þ F b22 R þ F rc4 R M rp þ M 4 ¼ 0 8P > < P F x ¼ F o2 px þ ðF b11 þ F b12 Þ cos hf þ F fc2 G3 sin a ¼ 0 F y ¼ F o2 py þ ðF b11 þ F b12 Þ sin hf þ Nf 2 G3 cos a ¼ 0 > :P Mo2 ¼ F b11 R þ F b12 R þ F fc2 R M fp þ M 2 ¼ 0
ð5Þ
ð6Þ
8P > < P F x ¼ F rc3 ðF b21 þ F b22 Þ cos hr F o3 x G4 sin a ¼ 0 F y ¼ Nr3 ðF b21 þ F b22 Þ sin hr F o3 y G4 cos a ¼ 0 > :P M o ¼ F b21 R F b22 R þ F rc3 R þ M3 ¼ 0 8P 3 F > < P x ¼ F fc1 ðF b11 þ F b12 Þ cos hf F o1 x G4 sin a ¼ 0 F y ¼ Nf 1 ðF b11 þ F b12 Þ sin hf F o1 y G4 cos a ¼ 0 > :P M o1 ¼ F b11 R F b12 R þ F fc1 R þ M1 ¼ 0
where M rp and M fp are the first outputs of each module to drive the rear and front active pulleys, respectively. F bij ði ¼ 1; 2; j ¼ 1; 2Þ is the force of belt acted on the rim of the rear active pulley and front active pulley; F o4 px and F o4 py are the interaction of forces between the robot body and the rear active pulley in x and y direction; F o2 px and F o2 py are the interaction forces between the robot body and front active pulley in x and y direction. M2 and M 4 are the rolling resistances, herein, M2 ¼ dNf 2 ; M 4 ¼ dN r4 ; d is the coefficient of rolling resistance. A force diagram relating to the front and rear frame is shown in Fig. 8. The equilibrium of two frames can be given by
8P F x ¼ F o4 fx þ F o3 x G2 sin a ¼ 0 > > >P < F y ¼ F o4 fy þ F o3 y G2 cos a ¼ 0 P > Mo4 ¼ F o3 x d3 sin hr þ F o3 y d3 cos hr Mrf > > : G2 d4 cos a cos hr þ G2 d4 sin a sin hr ¼ 0 8P F ¼ F o2 fx þ F o1 x G2 sin a ¼ 0 x > > > M o2 ¼ F o1 x d3 sin hf þ F o1 y d3 cos hf M ff > > : G2 d4 cos a cos hf þ G2 d4 sin a sin hf ¼ 0
ð7Þ
8P F x ¼ F o4 px F o4 fx F o2 px F o2 fx G1 sin a ¼ 0 > > > < P F ¼ F y o4 py F o4 fy F o2 py F o2 fy G1 cos a ¼ 0 P > > > Mo4 ¼ Mrp þ M rf þ M fp þ Mff G1 d2 cosða þ bÞ : þðF o2 fx þ F o2 px Þd1 sin b ðF o2 fy þ F o2 py Þd1 cos b ¼ 0
ð8Þ
Fb11
o2
Fo1 y
Fb 22 Fb 21
o3
Nr3
Fb12
Fo3 x
G4
G4
N f1
M 3 Frc 3 Fig. 9. Force diagram of the rear and front passive pulleys.
M rp
Fo4 px Fo4 fx
Fo4 py Fo4 fy
M fp
M rf
o4
M ff G1
ð10Þ
where N r3 and Nf 1 are the normal forces acted on the rear and front passive pulley from the ground; F rc3 and F fc1 are the tangential frictions exerted on the rear and front passive pulley; M3 and M 1 are the rolling resistances on the rear passive and front passive pulleys, respectively, herein, M3 ¼ dNr3 ; M 1 ¼ dN f 1 ; d is the coefficient of rolling resistance. The following part is the robot body, which is shown in Fig. 10. The following equilibrium about the robot body can be given by
o4
Fo3 y
ð9Þ
Fo2 px Fo2 fx
o2
Fo2 py
Fo2 fy
o3 Fig. 10. Force diagram of the robot body.
o1
o1
Fo1 x Ffc1 M1
ð11Þ
287
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Config. 5
Config. 1
10
10
Nr3 Nr 4 Nf
6 4
N f = N f1 + N f 2
Force (N)
θ f (°)
20 40 60 80 100 120 140 160 180
θ f (°)
Config. 6
Config. 2 20
Nr3 Nr 4 Nf
Nr3 Nr 4 Nf
15
5 0
0
20 40 60 80 100 120 140 160 180
Force (N)
0
10
4
0
0
15
6
2
2
20
Nr3 Nr 4 Nf
8
Force (N)
Force (N)
8
(282, 0)
-5
10 5 0
(258, 0)
(312, 0)
-5
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
Fig. 11. Numerical results of normal reaction forces for different config. 1, 5, 2 and 6.
From the mechanism design stated in Section 2, the output to frame keeps a proportional relation with the output to the active pulley. It can be indicated by
M rf ¼ KMrp M ff ¼ KMfp
ð12Þ
where Mrf and Mrp are the outputs to the rear active pulley and frame, respectively; Mff and Mfp are the outputs to the front active pulley and frame, respectively. K is ratio of reducer ratios and its value is 7.5, as listed in Table 1. A group of basic equations for all configurations are presented above. The differences among these configurations, are the constraints on the normal force, the tangential force and the rolling resistance. For instance, to describe the ‘‘config. 10” in the static case, the following constraints should be taken into consideration:
8 a ¼ 0 Ground without incline > > > > > < h < 360 180 > f > > > > 0 < hr < 180 > > > < N ¼ 0; N ¼ 0 r3 f2 No contact of the rear passive > F rc3 ¼ 0; F fc2 ¼ 0 > > > > > pulley and the front active pulley with the ground > > > > > M3 ¼ 0; M 2 ¼ 0 > > No forward or backward motion : M1 ¼ 0; M 4 ¼ 0
ð13Þ
4.2. Realizable postures Based on the above equations, analysis for each configuration is conducted here. For each configuration, there are two cases: without moving velocity and with moving velocity. The case ‘‘Without Moving Velocity” is that the maximum static friction is not exceeded and there is no rolling resistance. The case ‘‘With Moving Velocity” means that pulleys are subject to sliding friction and rolling resistance concurrently. The coefficient of static friction be-
tween pulley and ground ls is set to 0.5 and the coefficient of rolling friction d is considered as 2 mm. 4.2.1. Without moving velocity When the robot stays on the horizontal ground without moving velocity, rolling resistance exerted on the front and rear pulleys can be ignored. Thus, the constraints for each configuration in this static case are
a ¼ 0 M 1 ¼ 0; M 2 ¼ 0 M 3 ¼ 0; M 4 ¼ 0
ð14Þ
From the numerical results of reaction forces shown in Fig. 11, the normal reaction forces from ground can always be maintained larger than 0 for ‘‘config. 1” and ‘‘config. 5”. Due to the fact that ‘‘config. 1” is geometrically symmetrical to ‘‘config. 7” while ‘‘config. 5” is symmetrical to ‘‘config. 3”, we can find that ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 5” and ‘‘config. 7” can be kept under the friction conditions. The results of these configurations are summarized in Fig. 12. As shown in Fig. 11, for ‘‘config. 2”, the rear normal reaction force N r3 is larger than 0 just when the front angle hf is in the range ð180 6 hf 6 282 Þ, otherwise N r3 is less than 0. Similarly, the rear normal reaction force N r3 is larger than 0 just when the front angle hf is in the range ð258 6 hf 6 312 Þ for ‘‘config. 6”. With regard to ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”, only part of the postures can be balanced, since ‘‘config. 2” is geometrically symmetrical to ‘‘config. 8” while ‘‘config. 6” is symmetrical to ‘‘config. 4”. The controllable postures are listed for ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8” in Table 3. Concerning ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11”, and ‘‘config. 12”, there is only one line to show that the posture which can be balanced is that hf and hr should hold an one–one relation. As shown in Fig. 12, the configuration in the area where the robot cannot be balanced has the trend to change to another stable posture. The symbol ‘‘!” denotes the transition through control-
288
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
2
1
δ =0
360
11
340 320
4
300
12
280 260 240 220
θ r (°)
4
8 5
200
6
180 160
9
140
7
120 100
10 3
3
80 60 40
0
2
1
20 0
20
40
60
80
100 120 140 160
180 200 220 240 260 280 300 320 340 360
θ f (°) : Possible transition through controlling the actuators cooperatively Fig. 12. Controllable static postures of the robot.
Table 3 Controllable range of hf and hr of the robot in two cases. Config.
Without moving velocity
1 2 3 4 5 6 7 8 9 10 11 12 *
o4
Controllable range of hf and hr
θr
Equilibrium Point Position 1
Position 2
o1
M ff
With moving velocity
hf
hr
hf
hr
0°–180° 180°–282° 0° 0° 0°–180° 258°–312° 180° 180° hf þ hr ¼ 180 g 1 ðhf ; hr Þ ¼ 0* g 3 ðhf ; hr Þ ¼ 0* g 5 ðhf ; hr Þ ¼ 0*
0° 0° 0°–180° 228°–282° 180° 180° 0°–180° 258°–360°
0°– 180° 222°–360° 0° 0° — 292°–360° — 180° — g 2 ðhf ; hr Þ ¼ 0* g 4 ðhf ; hr Þ ¼ 0* g 6 ðhf ; hr Þ ¼ 0*
0° 0° 54°–180° 272°–310° — 180° — 285°–360°
θf2 o3
θf1
o2
θf
Fig. 13. Stability analysis for ‘‘config. 11”.
The balance of ‘‘config. 1”, ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 5”, ‘‘config. 6”, ‘‘config. 7” and ‘‘config. 8” belongs to stable equilibrium while the balance of ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” belongs to unstable equilibrium.
g i ðhf ; hr Þ ði ¼ 1; 2; . . . ; 6Þ describes a function of their variables.
ling the front and rear actuator outputs cooperatively. For instance, for the point in the region that ð0 < hf ; hr < 180 ; 0 < hf þ hr < 180 Þ, the robot can change posture to the posture ð0 < hf < 180 ; hr ¼ 0 Þ or ðhf ¼ 0 ; 0 < hr < 180 Þ in control of two actuators. As shown in Fig. 13, for the given angle of rear frame hr , there is just one angle of front frame hf to match it. When the rear module is kept in the current position, a rotation torque M ff is inevitably exerted on the front frame. If there is a deviation that makes the front angle become hf 1 , the torque M ff cannot provide enough force to lift the front frame back to the equilibrium point hf . Also if there is a deviation that makes the front angle become hf 2 , the torque M ff will make the front frame accelerate to go far away from the equilibrium point hf . In a word, the current state in ‘‘config. 11” is unstable equilibrium. Using the same method, it can be easily found that the balance in ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” is unstable equilibrium.
4.2.2. With moving velocity If the robot moves with a velocity of v (for instance, v ¼ 13:6 mm/s), rolling friction should be exerted at the rim of pulley which contacts the ground. Thus, the constraint for each configuration in the movement case on a horizontal plane is
a ¼ 0
ð15Þ
The numerical results of this static analysis are shown in Fig. 14. ‘‘Config. 1”, can be kept under the friction conditions since all the normal reaction forces are larger than 0, as shown in Fig. 14a. For ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”, just part of the postures can be balanced to keep the normal reaction forces on the active and passive pulleys larger than 0, as shown in Fig. 14b, c, d, f, and h, respectively. The corresponding ranges are listed in Table 3. Regarding to ‘‘config. 5” and ‘‘config. 7”, since the rear normal reaction force N r3 for ‘‘config. 5” and the front normal reaction force N f 1 for ‘‘config. 7” are always less than 0, there does not exist any suitable posture for the robot to be generated. Concerning
289
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Config. 1
Config. 2
9
14
8
12 10
Force (N)
7
Force (N)
Nr3 Nr 4 Nf
6 5
8 6 4
Nr3 Nr 4 Nf
4 3
2
(222, 0)
0
2
-2 0
20 40 60 80 100 120 140 160 180
180 200 220 240 260 280 300 320 340 360
θ f (°)
θ f (°)
(a)
(b)
Config. 3
Config. 4
12
20
N f1 Nf2 Nr
Force (N)
8 6 4
15 10
Force (N)
10
5 0
2
(54, 0)
0
-5
-2 0
20 40
60 80 100 120 140 160 180
θ r (°)
θ r (°)
(d)
Config. 5
10 5
20
10 5 0
0
0
20 40 60
80 100 120 140 160 180
θ f (°)
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
(e)
(f)
Config. 7 14 12 10 8 6 4 2
Config. 8 20 15
N f1 Nf2 Nr
Force (N)
Force (N)
(292, 0)
-5
-5
0 -2 -4 -6
Nr3 Nr 4 Nf
15
Force (N)
Force (N)
Config. 6
Nr3 Nr 4 Nf
15
(310, 0)
-10 180 200 220 240 260 280 300 320 340 360
(c) 20
(272, 0) N f1 Nf2 Nr
10 5 0
(285, 0) N f1 Nf2 Nr
-5 -10 0
20
40 60 80 100 120 140 160 180
-15 180 200 220 240 260 280 300 320 340 360
θ r (°)
θ r (°)
(g)
(h)
Fig. 14. Numerical results of normal reaction forces for different config. 1–8 considering rolling resistance (for instance,
‘‘config. 9”, the robot cannot perform the posture since the configuration is impossible to keep balance. For ‘‘config. 10”, ‘‘config. 11”, and ‘‘config. 12”, there are postures that can be generated for one–one relation between hf and hr . As shown in Fig. 15, the posture in the region where the robot cannot be balanced has the trend to change to another stable posture. The symbol ‘‘!” stands for the transition through controlling
v ¼ 13:6 mm=sÞ.
the front and rear actuator outputs cooperatively. For instance, for the point in the area that ð0 < hf ; hr < 180 Þ, it can make transition to the postures ð0 < hf < 180 ; hr ¼ 0 Þ or ðhf ¼ 0 ; 54 < hr < 180 Þ through effective control. The balance of ‘‘config. 1”, ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8” belongs to stable equilibrium while the balance of ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” belongs to unstable equilibrium.
290
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
2
1
δ = 2mm
360 340 320
4
300
12
280
11
260
4
8
240
θ r (°)
220
6
5
200 180 160
10
140 120
7
9
100
3
3
80 60 40
1
20 0
0
20
40
60
2 80
100 120 140 160 180 200 220 240 260 280 300 320 340
360
θ f (°)
: Possible transition through controlling the actuators cooperatively Fig. 15. Controllable postures of the robot with movement (for example,
v ¼ 13:6 mm=sÞ.
2
1 360 340
12
320
4
300 280
11
260 240
θ r (°)
220
6
5
200
4
8
180 160 140
3
120
10
9
7
3
100 80 60
P2
40
0
0
20
2
1
P1
20
40
60
80
100 120 140 160
180 200 220 240 260 280 300 320 340
θ f (°)
360
: Controllable static postures of robot : Controllable postures of robot with movement Fig. 16. Posture transition of the robot.
4.3. Posture transition In Fig. 12, we see that the robot can change from one stable posture to another continuously. On the other hand, when the robot moves forwards with a certain velocity, the stable posture sometimes cannot be changed from one to another continuously since the range of stable postures is strictly limited, as shown in
Fig. 15. However, the robot can overcome the blind area through the static posture transition, as shown in Fig. 16. For example, the robot with a certain moving velocity, can overcome the blind area from P1 to P 2 , through stopping the forward motion to perform posture transition statically. Note that, the posture transition discussed here can be executed without proper control of the interaction between the front and rear crawler modules. To perform the
291
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
400
1
3
9
7
5
8
11
4
2
10
6
12
θf θr
350 300
2 8
θ (°)
250 200
7
150
10
6
12
4
5
9
3
100
11
1
50 0
0
50
100
150
200
250
300
350
400
450
500
t (s) Fig. 17. Experiment scenes and angle relation when the robot performs posture transition without moving velocity.
discontinuous posture transition like that from ‘‘config. 12” to ‘‘config. 2”, the interaction between two crawler modules must be effectively controlled. We will discuss this in our future studies. 5. Experiments In order to verify the quasi-static analysis above, experiments were conducted to control the robot to perform the postures for each configuration. The velocities of two actuators were controlled manually by using two potentiometers. The internal interaction between the two crawler modules causes the robot to perform several postures. 5.1. Without moving velocity When the robot stays on the horizontal ground without moving velocity, rolling resistance exerted on the front and rear pulleys can be neglected. As shown in Fig. 17, postures are consecutively performed from ‘‘config. 1” to ‘‘config. 12”. Same as the numerical results of simulation, ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 5” and ‘‘config. 7” are stable equilibrium and thus can be achieved continuously. Regarding to ‘‘config. 9” and ‘‘config. 12”, the robot should be kept symmetrically otherwise it will make a posture transition to another pos-
ture, breaking the balance. For ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8”, the angle hf or hr just can be kept in part of the range from 0° to 360°. Once the angle hf or hr is beyond the related range, the balance is broken so as to cause the posture transition of the robot. For ‘‘config. 10” and ‘‘config. 11”, the one–one relation should be kept strictly to perform the posture. 5.2. With moving velocity If the robot moves with a certain velocity of v, rolling friction should be exerted at the rim of pulley that contacts the ground. As shown in Fig. 18, six postures including ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 8”, ‘‘config. 4”, ‘‘config. 2” and ‘‘config. 6” can be realized in the case that the robot moves forward, since other postures are unstable equilibrium stated in Section 4. As with the analysis in Section 4, experiments also show that the robot can keep the front frame at any position in the range 0 < hf < 180 in ‘‘config. 1”. In order to realize posture 3, a static transition is deployed necessarily, where the rear frame of the robot first rotates from 0° to 180°, then the robot moves with the rear frame lifted at the angle hr through the internal interaction between the two crawler modules. Like ‘‘config. 3”, it is inevitable that the robot adopts a static transition for ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”.
292
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
v = 13.6 mm/s
1
2
3
4
5
6
400
5
θf θr
350 300
4
θ (°)
250
3
2
200
6
150 100
1
50 0
0
50
100
150
200
Motion state Static posture transition
t (s)
250
300
350
400
450
Fig. 18. Experiment scenes when the robot moves forwards ðv ¼ 13:6 mm=sÞ.
6. Conclusions This paper has presented a new crawler mechanism in which a planetary gear reducer is employed as the transmission to give two outputs just using one actuator. The switching between modes of locomotion happens autonomously according to the change of terrain. A tracked robot composed of two modules of the proposed crawler mechanism has also been introduced, whose postures can be generated through controlling the two actuators. Since this crawler-driven robot is an under-actuated system, quasi-static analysis has been conducted to find out all the realizable postures. From the presented results, it is known that the robot moving forward at a certain velocity realizes less postures than that in static case. Static posture transition can be employed for the robot maintaining forward motion to overcome the blind spot of the posture transition. Experiments have also been executed to verify the quasi-static analysis.
References [1] Berkemeier MD, Poulson E, Groethe T. Elementary mechanical analysis of obstacle crossing for wheeled vehicles. In: Proceedings of 2008 international conference on robotics and automation, Pasadena, CA, USA; 2008. p. 2319—24. [2] Hirose S. Mechanical design of a mobile robot for external environments. J Robot Soc Jpn 2000;18(7):2–6.
[3] Moore E, Campbell D, Grimminger F, Buehler M. Reliable stair climbing in the simple hexpapod ‘rhex’. In: Proceedings of IEEE international conference on robotics and automation, Washington, DC; 2002. p. 2222–7. [4] Yoneda K, Ota Y, Ito F, Hirose S. Quadruped walking robot with reduced degrees of freedom. J Robot Mechatron 2001;13(2):190–7. [5] Jahanian O, Karimi G. Locomotion systems in robotic application. In: Proceedings of IEEE international conference on robotics and biomimetics, Kunming, China; 2006. p. 689–96. [6] Wang W, Du Z, Sun L. Dynamic load effect on tracked robot obstacle performance. In: Proceedings of the international conference on mechatronics, Kumamoto, Japan; 2007. p. 1–6. [7] Takayama T, Hirose S. Development of souryu i, ii-connected crawler vehicle for inspection of narrow and winding space. J Robot Mechatron 2003;15(1):61–9. [8] Mourikis AI, Trawny N, Roumeliotis SI, Helmick DM, Matthies L. Autonomous stair climbing for tracked vehicles. Int J Robot Res 2007;26(7):737–58. [9] Ben-Tzvi P, Goldenberg AA, Zu JW. Design, simulations and optimization of a tracked mobile robot manipulator with hybrid locomotion and manipulator capabilities. In: Proceedings of 2008 international conference on robotics and automation, Pasadena, CA, USA; 2008. p. 2307–12. [10] Lan G, Ma S, Inoue K, Hamamatsu Y. Development of a novel crawler mechanism with polymorphic locomotion. Int J Adv Robot 2007;21(3– 4):421–40. [11] Lan G, Ma S. Step-climbing analysis of a novel tracked robot. In: Proceedings of the 2005 IEEE international conference on robotics and biomimetics, Hong Kong and Macau; 2005. p. 544–9. [12] Quan Q, Ma S, Deng Z. Impact analysis of a dual-crawler-driven robot. Int J Adv Robot 2009;23(12–13):1779–97. [13] Ma S, Quan Q, Liu R. Posture analysis of a dual-crawler-driven robot. In: Proceedings of IEEE/ASME international conference on advanced intelligent mechatronics, Xi’an, China; 2008. p. 365–70. [14] Lan G, Ma S. Quasi-static analysis of a novel crawler-driven robot motion. J Robot Mechatron 2006;18(5):556–63.
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Controllable postures of a dual-crawler-driven robot Qiquan Quan, Shugen Ma * Department of Robotics, Ritsumeikan University, Japan
a r t i c l e
i n f o
Article history: Received 5 August 2009 Accepted 5 January 2010
Keywords: Tracked robot Crawler mechanism Quasi-static analysis Realizable postures Posture transition
a b s t r a c t This paper presents a tracked robot equipped with the proposed crawler mechanism, in which a planetary gear reducer is employed as a transmission device and provides two outputs in different forms with only one actuator. A robot which contains two crawler modules can generate several configurations through cooperatively controlling the two actuators. This tracked robot, which uses two actuators to give four outputs, however could have less realizable postures than that where each output is provided by one actuator exactly. To figure out what postures can be generated by the introduced dual-crawler robot, quasi-static analysis of the robot has been conducted while taking the rolling resistance into consideration and its realizable postures have been obtained numerically. The posture transition of the robot is also discussed in this paper. Experiments are conducted to verify the quasi-static analysis for each configuration. Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction Development of an efficient mobile mechanism for robots that are required to operate in irregular environments is always an important task. Robots using traditional wheeled mobile mechanisms can be programmed to travel over relatively smooth terrain easily; however, mobility over rugged terrain is limited by the diameter of wheels of robot [1,2]. Robots using legged mobile mechanisms can move well on uneven terrain, but they encounter several challenges, including difficulty of control and lack of stability [3,4]. Since tracked mobile mechanisms have advantages, like excellent stability, low pressure to terrain and simplicity of control, they have been widely deployed in irregular environments. Tracked mobile mechanisms, nevertheless, are still somewhat limited due to some mechanism parameters, such as the diameter of the front sprocket [5,6]. The most common way to improve the mobility and adaptability of tracked mobile mechanisms is to build a multi-track robot by linking several active or passive units in serial or parallel way [7–9]. However, to provide assisting actions and control the system correctly, it is necessary to add some extra actuators, mechanisms and control elements. In other words, these assisting actions cannot be performed autonomously. To resolve the difficulties outlined above, we have proposed a crawler mechanism with polymorphic locomotion [10,11]. This mechanism, which is equipped with a planetary gear reducer, makes use of only one actuator to provide two outputs. By determining the reduction ratio of two outputs in a suitable proportion, * Corresponding author. E-mail addresses: [email protected] (Q. Quan), [email protected]. ac.jp (S. Ma).
the crawler is capable of switching autonomously between two locomotion modes according to the terrain. The main characteristic of the mechanism is that the polymorphic locomotion is provided by one actuator and switching between the two modes of locomotion occurs autonomously. In this paper, we will introduce a tracked robot comprised of two modules of this crawler mechanism. The introduced tracked robot can generate several configurations by controlling properly the two actuators as well as each module’s polymorphic motions. However, this tracked robot, which employs two actuators to provide four outputs, would have less realizable postures than that where each output is produced by an individual actuator. This robot still has an advantage in decreasing the impact to actuators. When a robot moves over rough terrain, it inevitably collides with various obstacles, creating an impact effect on the driving actuator. When there is a collision transmitted from one output, another output can release part of the impact energy in our crawler mechanism. This impact absorption of the mechanism makes the actuator subject to less impact, and thus more safe [12]. The posture analysis of the dual-crawler-driven robot has been conducted without considering rolling resistance and did not discuss the posture transition [13]. This paper will give a systematical explanation of the controllable postures in both cases: ‘‘without moving velocity” and ‘‘with moving velocity”, and discuss the related posture transition. This paper presents static analysis of the robot to find out all the realizable postures and their possible transitions. The paper is organized as follows. The proposed crawler mechanism with polymorphic motion is introduced in Section 2 and the dual-crawler-driven robot is also given in this section. Section 3 gives all the possible geometric postures of this robot and the relevant geometric relations. Section 4 conducts the static analysis of the
0957-4158/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2010.01.001
282
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
dual-crawler-driven robot and discusses its realizable postures and posture transitions. Section 5 concludes the paper. 2. Mechanism of a dual-crawler robot By using two modules of the proposed crawler mechanism, a tracked robot with polymorphic locomotion was designed. In this section, we will first briefly describe the proposed crawler mechanism and then introduce the tracked robot consisting of two modules of this crawler mechanism. 2.1. Mechanism of a crawler module The proposed crawler mechanism is capable of providing two kinds of output with just one actuator. The first output is transmitted to the crawler-belt and drives the crawler to move forwards; the second one is employed to drive the connecting frame that links two sprockets of the crawler, as shown in Fig. 1. The planetary gear reducer was adopted as the main power transmission for our crawler mechanism, as shown in Fig. 1a. The input torque of the actuator is transmitted to the sun gear of the planetary gear reducer through a pair of bevel gears. Since the carrier of the planetary gear reducer is linked with an active pulley, the torque is derived from the sun gear and transmitted to the active pulley, the crawler belt, and acts as the first output to drive the crawler mechanism to forward or backward on even ground or slopes. As the second output, the torque is derived from the sun gear and transmitted to the ring gear of the planetary gear reducer, and then to a triangular gear reducer, and then to the connecting frame. The triangular reducer consists of three pairs of spur gears. The rotation of the connecting frame drives the crawler unit to rotate wholly around the input axle. 2.2. Polymorphic locomotion modes To describe the whole locomotion process while the crawler moves in irregular environment, we present three locomotion modes, referred to as ‘‘motion mode”, ‘‘rotation mode” and ‘‘recovering mode”.
(1) Motion mode (1, 2, 6 in Fig. 2): The crawler mechanism moves on an even terrain or slope like a normal tracked vehicle since the power of the actuator is transmitted to crawler-belt. (2) Rotation mode (3, 4 in Fig. 2): When the crawler mechanism contacts an obstacle, since the rotation of the crawler belt is stopped by the resistance from the ground and the power has to be transmitted to the connecting frame, the rotation of connecting frame drives the crawler mechanism to climb over the obstacle. (3) Recovering mode (5 in Fig. 2): Once the crawler mechanism has climbed over the obstacle, the power is transmitted to the connecting frame and drives the crawler mechanism to return back continuously until it recovers to the initial position. To achieve the proposed locomotion autonomously in irregular environments, the power transmission of the crawler should be designed to meet the following three conditions: (1) One motor input gives two outputs in the transmission. (2) The two outputs must rotate in the same direction. (3) The two reducer ratios are selected in a certain range. The mechanism of our crawler model equipped with a planetary gear reducer meets condition (1) and (2). Concerning the most important condition (2), we can determine the proportion of reduction ratios of two outputs within a certain range. The reduction ratios on outputs 1 and 2 have been designed to be 4 and 30, and the two outputs have the same rotation direction [10,14]. In motion mode, to drive the crawler mechanism to move on even ground or slope like a normal tracked vehicle, propulsion on the crawler belt has been designed larger than motion resistance. At the same time, rotation torque on the connecting frame is smaller than the rotation resistance generated by gravity of crawler and payload. In the same way, when the crawler mechanism contacts an obstacle, to climb over the obstacle in the proposed locomotion mode instead of track-slipping, the propulsion on the crawler belt has been designed smaller than the friction resistance so that the
Fig. 1. Mechanism of one crawler module.
283
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Belt
3
2
1
Frame 6
5
4
Fig. 2. Locomotion modes including ‘‘motion mode”, ‘‘rotation mode”, and ‘‘recovering mode” for step climbing.
Table 1 Physical parameters of the robot. Parameter
Fig. 3. Prototype of a dual-crawler-driven robot.
crawler belt can be fixed. Concurrently, the rotation torque is larger than the rotation resistance to lift vehicle body to climb over the obstacle. After the crawler mechanism has climbed up the obstacle, it can recover to the initial position autonomously. 2.3. Prototype of a dual-crawler-driven robot A tracked robot that is realized by connecting two modules of crawler mechanism through the body is shown in Fig. 3. This robot can not only switch autonomously between different locomotion
y
Rear
d3
θr
d4
o4 x
Weight of robot body Weight of frame Weight of active pulley Weight of passive pulley Distance between driven axles of two crawlers Distance between axles of rear active pulley and Center of Gravity (CG) of the body Distance between axles of active and passive pulleys Distance between axle of active pulley and CG of the frame Radius of pulley Radius of sun gear in planetary gear reducer Radius of planetary gear in planetary gear reducer Radius of ring gear in planetary gear reducer Ratio of reducer from motor to pulley Ratio of reducer from motor to triangle gear reducer Ratio of triangle gear reducer Ratio of reducer from motor to frame, i2 ¼ i21 i22 Ratio of reducer ratios ði2 ð30Þ=i1 ð4ÞÞ
β
Unit
G1 G2 G3 G4 d1 d2
1.06 0.14 0.14 0.14 190 95
kg kg kg kg mm mm
d3 d4
65 32.5
mm mm
R r1 r2 r3 i1 i21 i22 i2 K
29 4 4 12 4 3 10 30 7.5
mm mm mm mm
modes adapting to the terrain, but also generate several postures through controlling the two actuators cooperatively. The dimensions of the robot are shown in Fig. 4 and its physical parameters, which will be used in the following analysis are listed in Table 1.
d1
d2
Value
θf
Front
o2
o1
o3 Fig. 4. Dimensions of the dual-crawler-driven robot.
284
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292 Table 2 Incline angle b of robot body for each region of configurations.
θ r = 90°
hr
θ f = 270°
hf [0°, 180°]
[180°, 360°]
[0°, 180°]
b¼0
b ¼ arcsin
[180°, 360°]
hr b ¼ arcsin d3 sin d1
b¼
d3 sin hf d1 d sin hr d3 sin hf arcsin 3 d1
θ f = 90° θ f , θ r = 0°
Fig. 5 shows a sphere to represent the region of all configurations of the robot, where each corresponding configuration is on the surface of the sphere. For easier view, Fig. 6 shows all the configurations in an unfolded plane, where different combinations of the front angle hf and the rear angle hr can generate the relevant geometric possible postures. In total, there are 12 possible typical configurations when both the front angle hf and the rear angle hr vary from 0° to 360°. The different values of incline angle b can be classified into four groups shown in Figs. 5 and 6: Region rð0 6 hf 6 180 ; 0 6 hr 6 180 Þ, Region sð180 6 hf 6 360 ; 0 6 hr 6 180 Þ, Region tð0 6 hf 6 180 ; 180 6 hr 6 360 Þ, and Region uð180 6 hf 6 360 ; 180 6 hr 6 360 Þ. In the coordinate system ðxo4 yÞ of Fig. 4, the coordinates of the center of the rear passive pulley o3 , the center of the front active pulley o2 , the center of the front passive pulley o1 can be expressed as
θ r = 270° Fig. 5. Regions of all the configurations of a dual-crawler-driven robot represented by the latitude and longitude of a sphere.
3. Possible geometric configurations of the robot As shown in Fig. 4, the dual-crawler-driven robot consists of the rear crawler module, the robot body, and the front crawler module. The action that the front module and the rear module keep at different positions, can produce different configurations. Thus, there are several geometric possible postures for this dual-crawler robot. In order to define the configurations of this robot in a twodimensional environment, an orthogonal coordinate system ðxo4 yÞ is established at the center of the rear active pulley, as shown in Fig. 4. The x axis is parallel to the ground surface while the y axis is normal to the ground. The rear angle hr is the angle that the rear crawler module rotates from the x axis to the connecting frame of the rear module; the front angle hf is the angle that the front crawler module rotates from x axis to the connecting frame of the front module. As shown in Fig. 5, hf and hr are selected as the latitude and longitude, respectively, so that the point on the surface of the sphere can represent the corresponding configuration.
xo3 ¼ d3 cos hr ;
yo3 ¼ d3 sin hr
xo2 ¼ d1 cos b;
yo2 ¼ d1 sin b
xo1 ¼ d1 cos b þ d3 cos hf ;
ð1Þ
yo1 ¼ d1 sin b þ d3 sin hf
360
θ r (°)
(1) For the configurations in the Region r, the incline angle of the robot body b is always kept at 0°. (2) For the configurations in the Region s, the rear active pulley maintains full contact with ground surface while the front crawler module is lifted up. Since the centers of o1 and o4 have the same y coordinate element, the incline angle b can thus be described by
12
11 8
4
6
180
5
10
9 3
7
1 0
2 180 Fig. 6. All geometrical configurations of the robot.
360 θ f (°)
285
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
b ¼ arcsinðd3 sin hf =d1 Þ
ð2Þ
(3) For the configurations in the Region t, the front active pulley always stays in contact with ground surface while the rear crawler module is lifted up. The fact that centers of o2 and o3 have the same y coordinate element makes the incline angle b given by
b ¼ arcsinðd3 sin hr =d1 Þ
ð3Þ
(4) For the configurations in the Region u, both the front crawler module and the rear crawler module are lifted up. Because of the fact that centers of o1 and o3 have the same y coordinate element, the incline angle b can be described by
b ¼ arcsinððd3 sin hr d3 sin hf Þ=d1 Þ
ð4Þ
Table 2 summarizes the angle of incline of the robot body b for each different case of configurations. From the above analysis, we can obviously know that the robot can realize several possible configurations. For executing some tasks in rough terrain, the posture of robot body is desired to change according to the task. For example, if a manipulator is mounted on the robot body, as the base of the manipulator, the robot body can help the end-effector to perform the desired tasks. Thus, all the controllable postures should be found in the following analysis. 4. Quasi-static analysis of all configurations of the robot From the geometrical analysis of the dual-crawler-driven robot stated in Section 3, it is known that this robot could generate several configurations through cooperatively controlling the two actuators. However, this tracked robot, which makes use of two
Fo4 py o4
Fb 22
actuators to provide four outputs, could have less realizable postures than that where each output is given by only one individual actuator. In this section, we will conduct the quasi-static analysis of the robot and discuss realizable postures and their transitions. Since there are 12 typical configurations as stated in Section 3, there should be 12 different groups of equations for the statics. In order to get a group of statics equations, which can be employed to describe each posture clearly, we will present a group of basic formulations for the statics analysis.
4.1. Basic statics formulations for all configurations It can be easily found that the difference of each configuration is just that for each module of the robot, the active pulley contacts ground, the passive pulley contacts the ground, or both the active and passive pulleys contact the ground. To describe all the possible configurations in a general form, configuration 12 has been selected to be the basic posture for the general form. We presume that the normal forces and friction are exerted at the rim of each pulley. Thus, this general form can be used to describe all the possible configurations. The difference is that the normal force and frictions at the rim of pulley should be regarded as zero with respect to the different configurations. A slope, which is denoted by a, is also considered in the general form of equations, thereby this general form can describe all the possible configurations in a two-dimensional environment. Still, in the coordinate system, the directions of x axis and y axis are selected to be parallel and perpendicular to the slope, respectively. As shown in Fig. 7, the normal force N r4 and N f 2 , which are normal to the slope, are assumed to be exerted at the rim of the rear active and front active pulleys, respectively. Friction F rc4 and F fc2 , which are parallel to the slope, are also assumed to be exerted at the rim of the rear active and front active pulleys, respectively.
Fo2 py M fp
M rp Fo4 px
Fb11 o2 G3
M4 Frc 4
G3
Ffc 2
Fb 21
Nr 4
Nf2
Fo2 px M2
Fb12
o1
o3 α Fig. 7. Force diagram of the rear and front active pulleys.
Fo4 fy
Fo2 fy Fo4 fx
o4
Fo2 fx
o2 M rf o3
Fo3 x
M ff G2
Fo1 x
o1 Fo1 y
G2 Fo3 y Fig. 8. Force diagram of the rear and front frames.
286
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
where Mrf and Mff are the second outputs of each module to drive the rear and front frame, respectively. F o3 x and F o3 y are the interaction forces between the rear frame and rear passive pulley in x and y direction. F o4 fx and F o4 fy are the interaction forces between the rear frame and robot body in x and y direction. F o1 x and F o1 y are the interaction forces between the front frame and front passive pulley in x and y direction. F o2 fx and F o2 fy are the interaction forces between the front frame and robot body in x and y direction. A force diagram concerning the front and rear passive pulley is shown in Fig. 9. The equilibrium of the two passive pulleys can be obtained by
Rolling resistance M 4 and M 2 are assumed to be exerted likewise at the rim of front and rear active pulleys. As shown in Fig. 7, additional forces including normal force and friction are exerted at the rim of the rear active and front active pulleys, respectively. The equilibrium of the rear active and front active pulleys can be expressed as
8P > < P F x ¼ F o4 px þ ðF b21 þ F b22 Þ cos hr þ F rc4 G3 sin a ¼ 0 F y ¼ F o4 py þ ðF b21 þ F b22 Þ sin hr þ Nr4 G3 cos a ¼ 0 > :P Mo4 ¼ F b21 R þ F b22 R þ F rc4 R M rp þ M 4 ¼ 0 8P > < P F x ¼ F o2 px þ ðF b11 þ F b12 Þ cos hf þ F fc2 G3 sin a ¼ 0 F y ¼ F o2 py þ ðF b11 þ F b12 Þ sin hf þ Nf 2 G3 cos a ¼ 0 > :P Mo2 ¼ F b11 R þ F b12 R þ F fc2 R M fp þ M 2 ¼ 0
ð5Þ
ð6Þ
8P > < P F x ¼ F rc3 ðF b21 þ F b22 Þ cos hr F o3 x G4 sin a ¼ 0 F y ¼ Nr3 ðF b21 þ F b22 Þ sin hr F o3 y G4 cos a ¼ 0 > :P M o ¼ F b21 R F b22 R þ F rc3 R þ M3 ¼ 0 8P 3 F > < P x ¼ F fc1 ðF b11 þ F b12 Þ cos hf F o1 x G4 sin a ¼ 0 F y ¼ Nf 1 ðF b11 þ F b12 Þ sin hf F o1 y G4 cos a ¼ 0 > :P M o1 ¼ F b11 R F b12 R þ F fc1 R þ M1 ¼ 0
where M rp and M fp are the first outputs of each module to drive the rear and front active pulleys, respectively. F bij ði ¼ 1; 2; j ¼ 1; 2Þ is the force of belt acted on the rim of the rear active pulley and front active pulley; F o4 px and F o4 py are the interaction of forces between the robot body and the rear active pulley in x and y direction; F o2 px and F o2 py are the interaction forces between the robot body and front active pulley in x and y direction. M2 and M 4 are the rolling resistances, herein, M2 ¼ dNf 2 ; M 4 ¼ dN r4 ; d is the coefficient of rolling resistance. A force diagram relating to the front and rear frame is shown in Fig. 8. The equilibrium of two frames can be given by
8P F x ¼ F o4 fx þ F o3 x G2 sin a ¼ 0 > > >P < F y ¼ F o4 fy þ F o3 y G2 cos a ¼ 0 P > Mo4 ¼ F o3 x d3 sin hr þ F o3 y d3 cos hr Mrf > > : G2 d4 cos a cos hr þ G2 d4 sin a sin hr ¼ 0 8P F ¼ F o2 fx þ F o1 x G2 sin a ¼ 0 x > > >
ð7Þ
8P F x ¼ F o4 px F o4 fx F o2 px F o2 fx G1 sin a ¼ 0 > > > < P F ¼ F y o4 py F o4 fy F o2 py F o2 fy G1 cos a ¼ 0 P > > > Mo4 ¼ Mrp þ M rf þ M fp þ Mff G1 d2 cosða þ bÞ : þðF o2 fx þ F o2 px Þd1 sin b ðF o2 fy þ F o2 py Þd1 cos b ¼ 0
ð8Þ
Fb11
o2
Fo1 y
Fb 22 Fb 21
o3
Nr3
Fb12
Fo3 x
G4
G4
N f1
M 3 Frc 3 Fig. 9. Force diagram of the rear and front passive pulleys.
M rp
Fo4 px Fo4 fx
Fo4 py Fo4 fy
M fp
M rf
o4
M ff G1
ð10Þ
where N r3 and Nf 1 are the normal forces acted on the rear and front passive pulley from the ground; F rc3 and F fc1 are the tangential frictions exerted on the rear and front passive pulley; M3 and M 1 are the rolling resistances on the rear passive and front passive pulleys, respectively, herein, M3 ¼ dNr3 ; M 1 ¼ dN f 1 ; d is the coefficient of rolling resistance. The following part is the robot body, which is shown in Fig. 10. The following equilibrium about the robot body can be given by
o4
Fo3 y
ð9Þ
Fo2 px Fo2 fx
o2
Fo2 py
Fo2 fy
o3 Fig. 10. Force diagram of the robot body.
o1
o1
Fo1 x Ffc1 M1
ð11Þ
287
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Config. 5
Config. 1
10
10
Nr3 Nr 4 Nf
6 4
N f = N f1 + N f 2
Force (N)
θ f (°)
20 40 60 80 100 120 140 160 180
θ f (°)
Config. 6
Config. 2 20
Nr3 Nr 4 Nf
Nr3 Nr 4 Nf
15
5 0
0
20 40 60 80 100 120 140 160 180
Force (N)
0
10
4
0
0
15
6
2
2
20
Nr3 Nr 4 Nf
8
Force (N)
Force (N)
8
(282, 0)
-5
10 5 0
(258, 0)
(312, 0)
-5
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
Fig. 11. Numerical results of normal reaction forces for different config. 1, 5, 2 and 6.
From the mechanism design stated in Section 2, the output to frame keeps a proportional relation with the output to the active pulley. It can be indicated by
M rf ¼ KMrp M ff ¼ KMfp
ð12Þ
where Mrf and Mrp are the outputs to the rear active pulley and frame, respectively; Mff and Mfp are the outputs to the front active pulley and frame, respectively. K is ratio of reducer ratios and its value is 7.5, as listed in Table 1. A group of basic equations for all configurations are presented above. The differences among these configurations, are the constraints on the normal force, the tangential force and the rolling resistance. For instance, to describe the ‘‘config. 10” in the static case, the following constraints should be taken into consideration:
8 a ¼ 0 Ground without incline > > > > > < h < 360 180 > f > > > > 0 < hr < 180 > > > < N ¼ 0; N ¼ 0 r3 f2 No contact of the rear passive > F rc3 ¼ 0; F fc2 ¼ 0 > > > > > pulley and the front active pulley with the ground > > > > > M3 ¼ 0; M 2 ¼ 0 > > No forward or backward motion : M1 ¼ 0; M 4 ¼ 0
ð13Þ
4.2. Realizable postures Based on the above equations, analysis for each configuration is conducted here. For each configuration, there are two cases: without moving velocity and with moving velocity. The case ‘‘Without Moving Velocity” is that the maximum static friction is not exceeded and there is no rolling resistance. The case ‘‘With Moving Velocity” means that pulleys are subject to sliding friction and rolling resistance concurrently. The coefficient of static friction be-
tween pulley and ground ls is set to 0.5 and the coefficient of rolling friction d is considered as 2 mm. 4.2.1. Without moving velocity When the robot stays on the horizontal ground without moving velocity, rolling resistance exerted on the front and rear pulleys can be ignored. Thus, the constraints for each configuration in this static case are
a ¼ 0 M 1 ¼ 0; M 2 ¼ 0 M 3 ¼ 0; M 4 ¼ 0
ð14Þ
From the numerical results of reaction forces shown in Fig. 11, the normal reaction forces from ground can always be maintained larger than 0 for ‘‘config. 1” and ‘‘config. 5”. Due to the fact that ‘‘config. 1” is geometrically symmetrical to ‘‘config. 7” while ‘‘config. 5” is symmetrical to ‘‘config. 3”, we can find that ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 5” and ‘‘config. 7” can be kept under the friction conditions. The results of these configurations are summarized in Fig. 12. As shown in Fig. 11, for ‘‘config. 2”, the rear normal reaction force N r3 is larger than 0 just when the front angle hf is in the range ð180 6 hf 6 282 Þ, otherwise N r3 is less than 0. Similarly, the rear normal reaction force N r3 is larger than 0 just when the front angle hf is in the range ð258 6 hf 6 312 Þ for ‘‘config. 6”. With regard to ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”, only part of the postures can be balanced, since ‘‘config. 2” is geometrically symmetrical to ‘‘config. 8” while ‘‘config. 6” is symmetrical to ‘‘config. 4”. The controllable postures are listed for ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8” in Table 3. Concerning ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11”, and ‘‘config. 12”, there is only one line to show that the posture which can be balanced is that hf and hr should hold an one–one relation. As shown in Fig. 12, the configuration in the area where the robot cannot be balanced has the trend to change to another stable posture. The symbol ‘‘!” denotes the transition through control-
288
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
2
1
δ =0
360
11
340 320
4
300
12
280 260 240 220
θ r (°)
4
8 5
200
6
180 160
9
140
7
120 100
10 3
3
80 60 40
0
2
1
20 0
20
40
60
80
100 120 140 160
180 200 220 240 260 280 300 320 340 360
θ f (°) : Possible transition through controlling the actuators cooperatively Fig. 12. Controllable static postures of the robot.
Table 3 Controllable range of hf and hr of the robot in two cases. Config.
Without moving velocity
1 2 3 4 5 6 7 8 9 10 11 12 *
o4
Controllable range of hf and hr
θr
Equilibrium Point Position 1
Position 2
o1
M ff
With moving velocity
hf
hr
hf
hr
0°–180° 180°–282° 0° 0° 0°–180° 258°–312° 180° 180° hf þ hr ¼ 180 g 1 ðhf ; hr Þ ¼ 0* g 3 ðhf ; hr Þ ¼ 0* g 5 ðhf ; hr Þ ¼ 0*
0° 0° 0°–180° 228°–282° 180° 180° 0°–180° 258°–360°
0°– 180° 222°–360° 0° 0° — 292°–360° — 180° — g 2 ðhf ; hr Þ ¼ 0* g 4 ðhf ; hr Þ ¼ 0* g 6 ðhf ; hr Þ ¼ 0*
0° 0° 54°–180° 272°–310° — 180° — 285°–360°
θf2 o3
θf1
o2
θf
Fig. 13. Stability analysis for ‘‘config. 11”.
The balance of ‘‘config. 1”, ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 5”, ‘‘config. 6”, ‘‘config. 7” and ‘‘config. 8” belongs to stable equilibrium while the balance of ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” belongs to unstable equilibrium.
g i ðhf ; hr Þ ði ¼ 1; 2; . . . ; 6Þ describes a function of their variables.
ling the front and rear actuator outputs cooperatively. For instance, for the point in the region that ð0 < hf ; hr < 180 ; 0 < hf þ hr < 180 Þ, the robot can change posture to the posture ð0 < hf < 180 ; hr ¼ 0 Þ or ðhf ¼ 0 ; 0 < hr < 180 Þ in control of two actuators. As shown in Fig. 13, for the given angle of rear frame hr , there is just one angle of front frame hf to match it. When the rear module is kept in the current position, a rotation torque M ff is inevitably exerted on the front frame. If there is a deviation that makes the front angle become hf 1 , the torque M ff cannot provide enough force to lift the front frame back to the equilibrium point hf . Also if there is a deviation that makes the front angle become hf 2 , the torque M ff will make the front frame accelerate to go far away from the equilibrium point hf . In a word, the current state in ‘‘config. 11” is unstable equilibrium. Using the same method, it can be easily found that the balance in ‘‘config. 9”, ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” is unstable equilibrium.
4.2.2. With moving velocity If the robot moves with a velocity of v (for instance, v ¼ 13:6 mm/s), rolling friction should be exerted at the rim of pulley which contacts the ground. Thus, the constraint for each configuration in the movement case on a horizontal plane is
a ¼ 0
ð15Þ
The numerical results of this static analysis are shown in Fig. 14. ‘‘Config. 1”, can be kept under the friction conditions since all the normal reaction forces are larger than 0, as shown in Fig. 14a. For ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”, just part of the postures can be balanced to keep the normal reaction forces on the active and passive pulleys larger than 0, as shown in Fig. 14b, c, d, f, and h, respectively. The corresponding ranges are listed in Table 3. Regarding to ‘‘config. 5” and ‘‘config. 7”, since the rear normal reaction force N r3 for ‘‘config. 5” and the front normal reaction force N f 1 for ‘‘config. 7” are always less than 0, there does not exist any suitable posture for the robot to be generated. Concerning
289
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
Config. 1
Config. 2
9
14
8
12 10
Force (N)
7
Force (N)
Nr3 Nr 4 Nf
6 5
8 6 4
Nr3 Nr 4 Nf
4 3
2
(222, 0)
0
2
-2 0
20 40 60 80 100 120 140 160 180
180 200 220 240 260 280 300 320 340 360
θ f (°)
θ f (°)
(a)
(b)
Config. 3
Config. 4
12
20
N f1 Nf2 Nr
Force (N)
8 6 4
15 10
Force (N)
10
5 0
2
(54, 0)
0
-5
-2 0
20 40
60 80 100 120 140 160 180
θ r (°)
θ r (°)
(d)
Config. 5
10 5
20
10 5 0
0
0
20 40 60
80 100 120 140 160 180
θ f (°)
-10 180 200 220 240 260 280 300 320 340 360
θ f (°)
(e)
(f)
Config. 7 14 12 10 8 6 4 2
Config. 8 20 15
N f1 Nf2 Nr
Force (N)
Force (N)
(292, 0)
-5
-5
0 -2 -4 -6
Nr3 Nr 4 Nf
15
Force (N)
Force (N)
Config. 6
Nr3 Nr 4 Nf
15
(310, 0)
-10 180 200 220 240 260 280 300 320 340 360
(c) 20
(272, 0) N f1 Nf2 Nr
10 5 0
(285, 0) N f1 Nf2 Nr
-5 -10 0
20
40 60 80 100 120 140 160 180
-15 180 200 220 240 260 280 300 320 340 360
θ r (°)
θ r (°)
(g)
(h)
Fig. 14. Numerical results of normal reaction forces for different config. 1–8 considering rolling resistance (for instance,
‘‘config. 9”, the robot cannot perform the posture since the configuration is impossible to keep balance. For ‘‘config. 10”, ‘‘config. 11”, and ‘‘config. 12”, there are postures that can be generated for one–one relation between hf and hr . As shown in Fig. 15, the posture in the region where the robot cannot be balanced has the trend to change to another stable posture. The symbol ‘‘!” stands for the transition through controlling
v ¼ 13:6 mm=sÞ.
the front and rear actuator outputs cooperatively. For instance, for the point in the area that ð0 < hf ; hr < 180 Þ, it can make transition to the postures ð0 < hf < 180 ; hr ¼ 0 Þ or ðhf ¼ 0 ; 54 < hr < 180 Þ through effective control. The balance of ‘‘config. 1”, ‘‘config. 2”, ‘‘config. 3”, ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8” belongs to stable equilibrium while the balance of ‘‘config. 10”, ‘‘config. 11” and ‘‘config. 12” belongs to unstable equilibrium.
290
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
2
1
δ = 2mm
360 340 320
4
300
12
280
11
260
4
8
240
θ r (°)
220
6
5
200 180 160
10
140 120
7
9
100
3
3
80 60 40
1
20 0
0
20
40
60
2 80
100 120 140 160 180 200 220 240 260 280 300 320 340
360
θ f (°)
: Possible transition through controlling the actuators cooperatively Fig. 15. Controllable postures of the robot with movement (for example,
v ¼ 13:6 mm=sÞ.
2
1 360 340
12
320
4
300 280
11
260 240
θ r (°)
220
6
5
200
4
8
180 160 140
3
120
10
9
7
3
100 80 60
P2
40
0
0
20
2
1
P1
20
40
60
80
100 120 140 160
180 200 220 240 260 280 300 320 340
θ f (°)
360
: Controllable static postures of robot : Controllable postures of robot with movement Fig. 16. Posture transition of the robot.
4.3. Posture transition In Fig. 12, we see that the robot can change from one stable posture to another continuously. On the other hand, when the robot moves forwards with a certain velocity, the stable posture sometimes cannot be changed from one to another continuously since the range of stable postures is strictly limited, as shown in
Fig. 15. However, the robot can overcome the blind area through the static posture transition, as shown in Fig. 16. For example, the robot with a certain moving velocity, can overcome the blind area from P1 to P 2 , through stopping the forward motion to perform posture transition statically. Note that, the posture transition discussed here can be executed without proper control of the interaction between the front and rear crawler modules. To perform the
291
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
400
1
3
9
7
5
8
11
4
2
10
6
12
θf θr
350 300
2 8
θ (°)
250 200
7
150
10
6
12
4
5
9
3
100
11
1
50 0
0
50
100
150
200
250
300
350
400
450
500
t (s) Fig. 17. Experiment scenes and angle relation when the robot performs posture transition without moving velocity.
discontinuous posture transition like that from ‘‘config. 12” to ‘‘config. 2”, the interaction between two crawler modules must be effectively controlled. We will discuss this in our future studies. 5. Experiments In order to verify the quasi-static analysis above, experiments were conducted to control the robot to perform the postures for each configuration. The velocities of two actuators were controlled manually by using two potentiometers. The internal interaction between the two crawler modules causes the robot to perform several postures. 5.1. Without moving velocity When the robot stays on the horizontal ground without moving velocity, rolling resistance exerted on the front and rear pulleys can be neglected. As shown in Fig. 17, postures are consecutively performed from ‘‘config. 1” to ‘‘config. 12”. Same as the numerical results of simulation, ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 5” and ‘‘config. 7” are stable equilibrium and thus can be achieved continuously. Regarding to ‘‘config. 9” and ‘‘config. 12”, the robot should be kept symmetrically otherwise it will make a posture transition to another pos-
ture, breaking the balance. For ‘‘config. 4”, ‘‘config. 6” and ‘‘config. 8”, the angle hf or hr just can be kept in part of the range from 0° to 360°. Once the angle hf or hr is beyond the related range, the balance is broken so as to cause the posture transition of the robot. For ‘‘config. 10” and ‘‘config. 11”, the one–one relation should be kept strictly to perform the posture. 5.2. With moving velocity If the robot moves with a certain velocity of v, rolling friction should be exerted at the rim of pulley that contacts the ground. As shown in Fig. 18, six postures including ‘‘config. 1”, ‘‘config. 3”, ‘‘config. 8”, ‘‘config. 4”, ‘‘config. 2” and ‘‘config. 6” can be realized in the case that the robot moves forward, since other postures are unstable equilibrium stated in Section 4. As with the analysis in Section 4, experiments also show that the robot can keep the front frame at any position in the range 0 < hf < 180 in ‘‘config. 1”. In order to realize posture 3, a static transition is deployed necessarily, where the rear frame of the robot first rotates from 0° to 180°, then the robot moves with the rear frame lifted at the angle hr through the internal interaction between the two crawler modules. Like ‘‘config. 3”, it is inevitable that the robot adopts a static transition for ‘‘config. 2”, ‘‘config. 4”, ‘‘config. 6”, and ‘‘config. 8”.
292
Q. Quan, S. Ma / Mechatronics 20 (2010) 281–292
v = 13.6 mm/s
1
2
3
4
5
6
400
5
θf θr
350 300
4
θ (°)
250
3
2
200
6
150 100
1
50 0
0
50
100
150
200
Motion state Static posture transition
t (s)
250
300
350
400
450
Fig. 18. Experiment scenes when the robot moves forwards ðv ¼ 13:6 mm=sÞ.
6. Conclusions This paper has presented a new crawler mechanism in which a planetary gear reducer is employed as the transmission to give two outputs just using one actuator. The switching between modes of locomotion happens autonomously according to the change of terrain. A tracked robot composed of two modules of the proposed crawler mechanism has also been introduced, whose postures can be generated through controlling the two actuators. Since this crawler-driven robot is an under-actuated system, quasi-static analysis has been conducted to find out all the realizable postures. From the presented results, it is known that the robot moving forward at a certain velocity realizes less postures than that in static case. Static posture transition can be employed for the robot maintaining forward motion to overcome the blind spot of the posture transition. Experiments have also been executed to verify the quasi-static analysis.
References [1] Berkemeier MD, Poulson E, Groethe T. Elementary mechanical analysis of obstacle crossing for wheeled vehicles. In: Proceedings of 2008 international conference on robotics and automation, Pasadena, CA, USA; 2008. p. 2319—24. [2] Hirose S. Mechanical design of a mobile robot for external environments. J Robot Soc Jpn 2000;18(7):2–6.
[3] Moore E, Campbell D, Grimminger F, Buehler M. Reliable stair climbing in the simple hexpapod ‘rhex’. In: Proceedings of IEEE international conference on robotics and automation, Washington, DC; 2002. p. 2222–7. [4] Yoneda K, Ota Y, Ito F, Hirose S. Quadruped walking robot with reduced degrees of freedom. J Robot Mechatron 2001;13(2):190–7. [5] Jahanian O, Karimi G. Locomotion systems in robotic application. In: Proceedings of IEEE international conference on robotics and biomimetics, Kunming, China; 2006. p. 689–96. [6] Wang W, Du Z, Sun L. Dynamic load effect on tracked robot obstacle performance. In: Proceedings of the international conference on mechatronics, Kumamoto, Japan; 2007. p. 1–6. [7] Takayama T, Hirose S. Development of souryu i, ii-connected crawler vehicle for inspection of narrow and winding space. J Robot Mechatron 2003;15(1):61–9. [8] Mourikis AI, Trawny N, Roumeliotis SI, Helmick DM, Matthies L. Autonomous stair climbing for tracked vehicles. Int J Robot Res 2007;26(7):737–58. [9] Ben-Tzvi P, Goldenberg AA, Zu JW. Design, simulations and optimization of a tracked mobile robot manipulator with hybrid locomotion and manipulator capabilities. In: Proceedings of 2008 international conference on robotics and automation, Pasadena, CA, USA; 2008. p. 2307–12. [10] Lan G, Ma S, Inoue K, Hamamatsu Y. Development of a novel crawler mechanism with polymorphic locomotion. Int J Adv Robot 2007;21(3– 4):421–40. [11] Lan G, Ma S. Step-climbing analysis of a novel tracked robot. In: Proceedings of the 2005 IEEE international conference on robotics and biomimetics, Hong Kong and Macau; 2005. p. 544–9. [12] Quan Q, Ma S, Deng Z. Impact analysis of a dual-crawler-driven robot. Int J Adv Robot 2009;23(12–13):1779–97. [13] Ma S, Quan Q, Liu R. Posture analysis of a dual-crawler-driven robot. In: Proceedings of IEEE/ASME international conference on advanced intelligent mechatronics, Xi’an, China; 2008. p. 365–70. [14] Lan G, Ma S. Quasi-static analysis of a novel crawler-driven robot motion. J Robot Mechatron 2006;18(5):556–63.