- Email: [email protected]
Mechanical design of multifunctional quadruped
Mechanism and Machine Theory 38 (2003) 463–478 www.elsevier.com/locate/mechmt
Mechanical design of multifunctional quadruped T. Zielinska
a,b,*
, John Heng
a
a
b
NTU––Nanyang Technological University, Singapore Institute of Aircraft Engineering and Applied Mechanics, Warsaw University of Technology, UI. Nowowiejska 244, Warsaw 00 665, Poland
Received 10 July 2001; received in revised form 24 October 2002; accepted 8 November 2002
Abstract This paper describes the mechanical design of a multifunctional four-legged walking machine that is being developed at the Robotics Research Centre, NTU. The major factors influencing the design requirements include minimisation of the weight of the machine, large workspace of the legs, good energy efficiency and relatively high walking speed. The designed walking machine can adopt a variety of configurations such as insect, mammalian, reptile, or human like. The design is invertable and the machine using the legs as manipulators can even perform basic pick and place functions. Lengths of machine leg segments are: 0.12 m––thigh segment and 0.14 m––shank segment, body has a square shape with the side 0.08 m. Total weight is 1.5–2 kG (depends on the type of feet attached and construction of the body frame). The speed of motion depends on the type of implemented gait (side walking with separated legs motion and body transfer, turning motion, stair climbing etc.), during basic straight line quadruped crawl (only one leg at a time in transfer) on a horizontal surface it is 0.045 m/s. It should be stressed that the speed depends on leg configuration and the step length. For shorter steps the number of leg transfers for a given travelling distance is larger and the walking speed is lower. On the other hand, a shorter step length implies that the leg resultant mass will be transferred over a shorter distance in relation to the body frame and the total displacement of the machine centre of mass will be smaller which increases the vehicle stability compared with the situation when the step is longer. Considering different gaits with diverse configurations we can summarise that the walking speed range of the legged autonomous vehicle machine is between 0.001 and 0.045 m/s. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Walking machine; Legged machine; Invertable quadruped; Quadruped design
*
Corresponding author. Address: Institute of Aircraft Engineering and Applied Mechanics, Warsaw University of Technology, UI. Nowowiejska 244, Warsaw 00 665, Poland. E-mail addresses: [email protected] (T. Zielinska), [email protected] (J. Heng). 0094-114X/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00004-1
464
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
1. Introduction There have been many publications underlying that legged locomotion is the best form of locomotion through varying terrain, as compared to using wheels. The surface of a terrain may be uneven, soft, muddy and generally unstructured. Legs in the biological environment demonstrate significant advantage in such situations. It is on this basis that the motive to design and develop a highly flexible walking machine to mimic our four-legged biological cousins begins. Designers of legged machines must take several decisions, which influences the technical features of these systems. As the most important of these, we can list: design of the mechanical structure and choice of leg configuration (choice of number of legs, their kinematics structure, joint design solution), specification of actuating and drive mechanisms (choice of motors type, evaluation of their power, design of motor placement and evaluation of methods of motion transmission from motors to the legs joints), evaluation of expected power consumption in relation to the machineÕs weight, payload, motion conditions (soft, hard terrain, inclined terrain, etc.) and assumed method of walk (speed of motion, number of legs supporting the body during walk, etc.). An important consideration is the adequate design of the control system (on board/ of board control system, control software, hardware and software control systems architecture, distribution of the on board utilities, cables, sensors which influences the stability conditions, etc.). Machines autonomy depends on the internal sensors delivering the information about the internal state of the device and on the external sensors detecting the external environmental conditions. This information must be properly used by the control software that finally determines the machineÕs ÔintelligenceÕ. Our paper deals with the problems of mechanical design which determines the walking machine posture, energy efficiency, range of walking speed and types of gaits which can be later implemented.
2. Overview of the most common legs design ideas Several basic configurations exist for walking machines that have been developed so far. Fig. 1 shows the schematic diagram of an insect leg type. This pattern is used in leg design of average size hexapods––[1–3]. (i.e. LAURON I: 12–15 kG weight, 0.6 m body length, 0.5 m body width, and Hamlet, Katharina, Masha with comparable parameters). Fig. 2 shows the pantograph design type. The points at A and B move linearly (usually driven by a lead screw motor) and if the mechanical linkage is designed carefully, point F can be made to closely follow a straight line (i.e. heavy ASV [4,5], Autopod 0.33, MECANT, Plustech, RIMMO, and average size lighter hexapod COMET [6], quadrupeds QUARUP [7] and TITAN [1]). Fig. 3 shows a leg configuration that is commonly used in a variety of light, small legged vehicles (i.e. [1,8,9]). This design employs two servo motors attached back to back and each motor is responsible for either the lift or swing axes. This design if employed carefully allows the design to be invertable [9].
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
465
Fig. 1. Leg structure typical for insects.
Fig. 4 shows a hybrid leg design which couples the pantograph technique with a linear actuator at its end to generate leg lift and leg swing motions (i.e. MELWALK, MELWALK MARK III [10]). Fig. 5 shows a 3 DOF leg design which employs linear actuation for the leg lift and linear (stretch) motions which employ a rotary actuator for the leg swing actions (i.e. AMBLER [11], DANTE, ROWER [1]). According to our knowledge, in current designs, individual motors are responsible for individual (or single) actuation of the leg joints.
Fig. 2. Pantograph type leg.
466
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 3. Typical two degree of freedom leg.
Fig. 4. Hybrid design.
Fig. 5. Leg with two prismatic joints.
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
467
3. Design considerations 3.1. Mechanical design It was proposed that a collective motor actuation design should be employed. This idea utilizes the collective effort of 2 motors to cooperatively control leg lift and swing action. Fig. 6 shows the current development state of the legged autonomous vehicle (LAVA) developed at the Robotics Research Centre, NTU, Singapore. To minimize the weight and increase energy output of each servo motor, each leg with its 3 DOF is driven through an inverse differential gear drive. Two motors are located in the hip section and the third is in the knee section. The hip motors work collectively to generate the combined leg swing and lift. The benefit of such a system is that the leg lift motor does not have to carry the leg swing motor as if it was a dead weight in conventional leg actuation design. Each of the servo motors is coupled through a worm gear system to ensure a stable system lock to allow the motors to power down when the legged vehicle merely needs to stand thus saving electrical power. Each individual leg section was carefully designed to provide individual lift and swing angles beyond 180° thus providing the vehicle with a high degree of motion dexterity to perform a variety of configurations beyond the basic walking and support functions. Special care was taken to ensure that minimal obstruction was encountered between the legs and the body when the leg were being driven through its large leg lift and sweep angles. Furthermore, minimal modifications would be required for LAVA to be waterproofed. A close up drawing of each individual leg is shown in Fig. 7. Each of the hip motors is rated at 4.5 W (MAXON motor) coupled through a 2 stage 19:1 planetary gear box and a 40:1 worm gear. The knee motor is rated at 3 W with a 3 stage 76:1 planetary gearbox and a 40:1 worm gear. All motors are coupled to a 16 counts per turn 2 phase TTL compatible magnet digital encoder. At an average motor input speed of 3850 rpm, the thigh swing and lift speeds are calculated to achieve 30°/s while achieving a combined torque (both motors working collectively) of up to 4 Nm. This figure includes the efficiency and friction losses at the worm and gearboxes. According to the manufacturer, the epicyclic gearbox has an 81% efficiency and the worm gear is estimated to have a 70% efficiency. The use of a differential gearbox for the thigh motors warrants a high degree of mechanical precision and tight tolerance (0.1 mm) to ensure a smooth and fluid action of the thigh motion
Fig. 6. The current development state of the LAVA.
468
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 7. Close up drawing of individual leg with differential drive.
Fig. 8. Different LAVA configuration; concept (a) actual implementation (b).
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
469
Fig. 8 (continued)
with minimal backlash and friction through it wide swing and lift action. Careful bearing selection, design and orientation were required to ensure that seizure, mechanical wear and heat build up was minimized.
470
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 9. LAVA Ôback to bellyÕ transfer.
3.2. Multifunctional quadruped capability The following figures show LAVA employed in a variety of configurations. In Fig. 8 the drawings on the left (a) shows the intended conceptual LAVA configuration and the corresponding photograph on the right (b) shows the actual implementation. LAVA has overturning capabilities which are useful for the exploration. Scientific instruments like probes, sensors etc. cannot be located symmetrically on the body of the machine as they usually are complex and expensive. If during a mission, a robot topples over and falls down with its back on the ground, it will be able to get up and return to the normal position. Fig. 9 illustrates the sequence of postures designed for LAVA Ôback to bellyÕ active transition [14], and Fig. 10 shows LAVA in one of those postures.
4. Discussion of advantage and disadvantages of proposed design 4.1. Differential mechanism and energy/torque demand LAVA leg actuation employs a differential gear drive and worm gears which offer a self-locking feature. So far as to the authorsÕ knowledge, this combined solution has not been implemented in
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
471
Fig. 10. Lava between posture 9 and 10.
any existing known walking machines. Mechanical locking of the leg in support phase has been proposed by Hirose [12] but many of the existing legged machines offer a back drivable actuation system. The self-locking feature contributed significantly in aiding the torque required during the leg support phase. Our goal was not only to attain good mechanical efficiency but also large leg work-space which offers the ability to walk with many different gaits and with different legs postures. Moreover, LAVA has an active over-turn ability. This property is exhibited only by the hexapod IOAN with tip-over by active motion. Such feature is useful when the device is used for exploration. Scientific instruments like probes, sensors etc. cannot be located symmetrically on the body of the vehicle as they are expensive. Fig. 11 shows the expected torque demand in quadruped crawl gait with Fig. 12 showing the angles related not to the joints but to the motors (angular motions in hip joints were recalculated
Fig. 11. ALV2 (2 kG) torque demand during quadruped crawl.
472
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 12. Motors motion during leg transfer phase. Angles are related to the expected angle contributed by motor rotation (h1 þ h2 for one hip motor, h1 h2 for second hip motor).
to the angles which were contributed by the motors through the differential mechanism). For simplicity in both drawings, the horizontal axis is expressed in scaled units of control time. In Fig. 11, one control step takes in reality of 60 ms. Hence the support phase takes 15 60 ms ¼ 900 ms. This duration of the support phase has been used in the majority of implemented gaits. In Fig. 12 one control step is 20 ms, so the transfer phase takes 300 ms. The torque was calculated using the Jacobian approach, the real values of reaction forces were considered as the input values (for horizontal terrain). In calculations, the torque required for supporting the body and for powering of the body motion were distinguished. The largest torque required (Fig. 11––line with circles) was for the support of the body. With an ideal self-locking mechanism in place, this torque demand can be neglected and (in hip joint) we need to consider only the torque consumption for machine motion. Assuming self-locking mechanism to be highly effective, we can conclude that if the motors are chosen to deliver driving torque, this torque instead of supporting the machine (and load) itself can be used to produce the machine motion with extra payload. In other words this torque can be used to compensate for a bigger load which demands bigger body motion powering torque. In classic legged vehicle design this is not achievable. In actual practice, the self-locking mechanism does not approach 100% effectiveness and moreover the body powering torque significantly increases when walking on uneven terrain or when walking by turning/side motion. Non-back drivable nature of the worm gear is evident in standing support, but it helps also to support the body during walking. In this situation the load distributed over the legs is not able to force the Ônon-usableÕ rotation of the worm gears and the motor shafts connected to them. We tested a family of GROVEN (GROVEN I, II, III) hexapods (35–45 kG) with legs designed similarly to the legs of LAVA [13]. In the case of the
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
473
first GROVEN prototype (45 kG) the design procedure neglected the self-locking capability of the worm gears and assumed 0 kG payload. In consequence the real (measured) payload of the machine was 40 kG in tripod gait. During estimation of the motor parameters, it is suggested that the demand for torque supporting the weight of the body cannot be fully ignored. We need to add a safety margin to cover the different walking conditions (despite of the margin which must be added to cover the limited mechanical efficiency and presence of friction). In our works, on the evaluation of motors for the next GROVEN prototypes, we did not add the torque covering the limited efficiency but instead we neglected the torque benefits resulting from the self-locking. The experiments with LAVAÕs different gaits over different surfaces (with the inclination up to 15°) showed that the motors and reduction ratios were selected properly. 4.2. Differential mechanism and walking machine speed demand For the discussion provided below we assumed that the reductions are equal in classic design and design with differential gear. In differential gear mechanisms, one motor moves following the sum of two angles at the hip joint (angle related to leg motion up/down plus angle related to the backward/forward leg-end motion). Another hip motor follows the difference of hip angles. As a result one of the hip motors (depends on anglesÕ sign) must move with a velocity higher than that expected in classic hip design. We noticed that in the implemented gait, one of the LAVA hip motors is expected to cover the angle change 2.7 times bigger (Fig. 12) than the angle increment in classical design where separate motors powers separately forward/backward and upward/downward thigh link motion. It means that the speed expected from motor in differential mechanism can be 2–3 times bigger than in classic design. For the same gear reduction classically designed machine will be about twice the speed of LAVA. Other factors which are limiting the body traveling speed is the number of leg steps which must be completed to cover the given body distance. If the leg step length (the range of leg-end backward shift in relation to the body in one support phase) is shorter, the number of steps over the distance increases thus increasing the time spent on ineffective transfer phase (raising up and moving down the leg). Hence the step length should be long. However a large increase in the step length over the evaluated limit will cause instability especially when walking on undulating or soft terrain.
5. Optimization of leg proportions considering the power demand The length of leg links was chosen after the study of total power demand (power demand for 3 motors) for one leg during the support phase. Seven listed further configurations were considered. For each configuration, the energy consumption in relation to the thigh/shank length proportion was calculated. For the final LAVA design, the leg links proportion was chosen such that this proportion was shown to have relatively low power demand for different leg configurations.
474
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
First the torque in the leg joints were calculated, using (as mentioned) the Jacobian approach [14]: s ¼ J ðhÞT F
ð1Þ
where J ðhÞT is the transposed Jacobian matrix [m]. F represents the force vector exerted by the leg to support and move the body. The components of the force vector are equal to reaction forces, but with opposite signs F ¼ ½fx ; fy ; fz T ½N
ð2Þ
Jacobian J ðhÞ is expressed in the same frame as the force F (base frame––x0 ; y0 ; z0 ), and s is the vector of joint load torques [Nm]. The direct and inverse kinematics problem was solved for the legs [9]. The typical Denavit–Hartenberg approach was applied. For the inverse problem the algebraic approach was applied. As this problem is not complex and kinematics of 3 DOF mechanism is very often described in the robotics books we will not go into details. The leg-end forces were provided for the solution of the so called force distribution problem which is well known for scientists working on walking machines (i.e. [15]). The expected hip motors power Phip was approximated by Phip ¼ ðs1 þ s2 ÞDh=ðts 360°Þ
ð3Þ
The reduction ratio minimizing the motor torque requirement was chosen paying the attention to walking speed. If this speed is equal to v, and step length is equal to s, the time ts of support phase is v=s. The biggest angular velocity is expected during leg transfer phase. This value strongly depends on the gait pattern and shape of the leg-end trajectory during transfer phase. In quadruped crawl gait with duty factor 0.75, the leg supports the ground by 75% of the gait period and only 25% of this period leg is in the transfer phase, which means (approximating) that angular speed in transfer must be at least 3 times bigger than that in support. We assumed that in the worst case, the transfer can be 4 times shorter than the support. This can also cover a bigger range of angle change in transfer due to the different shape of leg-end trajectory than that in support. From there, the joint angular speed is equal to h_hip ¼ 4Dhmax
hip =ts
ð4Þ
where (differential mechanism): Dhmax
hip
¼ maxfjDh1 þ Dh2 j; jDh1 Dh2 jg
The reduction evaluated for this speed is equal to the average motor speed divided by the joint angular speed. Average motor speed mv is expressed in rpm (rotation per minute), assuming that unit of time ts is also minute, the reduction ri is equal to rhip ¼ mv =ð4Dhmax
hip =ð360° ts ÞÞ
ð5Þ
Neglecting power (energy) benefit resulting from the self-locking, but not considering the limited mechanical efficiency, we can express that the expected hip motors torque is equal to
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
sm1 þ sm2 ¼ ðs1 þ s2 Þ=rhip
475
ð6Þ
Considering in formula (3) Eqs. (5) and (6) hip motors power Phip can be approximated as Phip ¼ ðsm1 þ sm2 Þmv
ð7Þ
In power demand analysis we used formula (1) for torque evaluation, next the gear ratio using (4) and (5) was calculated and from (6) and (7) motors power were evaluated. The knee motor power Pknee was calculated in similar way, the analysis of power demand takes into account the total torque P ¼ Phip þ Pknee . In the study of power demand the following configurations were considered––Fig. 11 (in a list below, initial value on the beginning of support phase is given) Configuration 1: Configuration 2: Configuration 3: Configuration 4: Configuration 5: Configuration 6: Configuration 7:
h1 ¼ 42°, h2 ¼ 45°, h3 ¼ 98° (insect configuration) h1 ¼ 42°, h2 ¼ 0°, h3 ¼ 90° (typical reptile configuration) h1 ¼ 42°, h2 ¼ 10°, h3 ¼ 80° (reptile configuration) h1 ¼ 42°, h2 ¼ 10°, h3 ¼ 92° (intermediate configuration: insect–reptile type) h1 ¼ 42°, h2 ¼ 20°, h3 ¼ 94° (insect configuration) h1 ¼ 42°, h2 ¼ 30°, h3 ¼ 50° (intermediate configuration: reptile–mammal type) h1 ¼ 42°, h2 ¼ 30°, h3 ¼ 96° (insect configuration)
As shown in Fig. 14, in many cases (i.e. configurations) a ratio with l1 =l2 lying within the range of 0.3–0.9 is associated with smaller motor power rather than that for other leg links proportions. For all the cases considered, insect Configuration 7 is optimal (low motor power requirements) for l1 =l2 greater than 0.6. Such a low power requirement can be observed in other configurations only for exactly fixed l1 =l2 (e.g. for Configuration 5, where l1 =l2 ¼ 0:3 or for Configuration 4 where l1 =l2 ¼ 0:48).
Fig. 13. General view of leg (insect configuration).
476
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 14. Influence of leg configuration and proportion on motor power.
Despite of the lower power demand for l1 =l2 greater than 0.5, we can notice that for l1 =l2 > 0:5 the power requirement for one configuration can be doubled when compared with another one. For example, for l1 =l2 equal to 1, the power requirement depends noticeably on the configuration and is in the range of 2 W (Configuration 7) to 3.8 W (Configuration 1). On the other hand, for the ratio smaller than 0.3, the range of power change in relation to configuration is much greater, e.g. 2.25–7.25 W for l1 =l2 ¼ 0:2. Summarising––for the designer––the ratio l1 =l2 between 0.6 and 0.9 (to 1.0) can be suggested as the optimum choice as for that ratio, in all legs configuration motor power is minimal or is not far from minimal demand. In the design of LAVA legs l1 =l2 ¼ 0:857 was chosen (see Fig. 13).
6. Conclusions The mechanical structure of the prototype must be very well assembled for effective selflocking. When this is the case, smaller torque in the hip is required and gear reduction can be decreased which increases walking speed. As the payload and speed are inversely proportional in relation to the gear reduction, differential gear has the advantage in torque when compared to the classic design (torque benefit depends on the quality of manufacturing). The expected gain in payload can even be doubled while speed reduction is less than 2 times (for equal motors and reductions)––which means that by decreasing the reduction gear by two in our prototype we can achieve similar payload and speed when compared with classic machines (with two time greater reduction).
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
477
For the better energy efficiency the leg link proportion and body size influencing the walking step length must be designed properly. The proposed novel leg design can be utilized in every multi-legged statically stable machine. The four legged machine LAVA has been built according to our design considerations. Multi-level real time based control system was developed and the prototype moves successfully using several types of gaits. [16] presents a detailed discussion of motor power evaluation considering the assumed vehicle mass and expected motion speed, [17] describes the LAVA control system and experiments with compliance position force control. The development and usability of walking machines can be constantly improved by the aid of proper design. The final goal of this presented work is to develop an autonomous walking machine, which will be optimised from the point of view of energy consumption and will be able to operate autonomously in natural conditions.
Acknowledgement The authors would like to express their thanks to Dr. G. Seet, Director of the Robotics Research Centre, NTU who supported the conducted research.
References [1] K. Berns, Walking Machines Catalogue (www.fzi.de/ids/WMC). [2] W. Ilg, K. Berns, A learning architecture based on reinforcement learning for adaptive control of walking machine LAURON, Robotics and Autonomous Systems 15 (1995) 321–334. [3] U. Muller-Wilm, H. Cruse, J. Eltze, J. Dean, H.-J. Weidemann, F. Pfeiffer, Kinematic model of a stick insect as an example of a six-legged walking system, Adaptive Behavior 1 (2) (1992) 1–15. [4] K.J. Waldron et al., Configuration design of the adaptive suspension vehicle, Robotics Research 3 (2) (1984) 37–48. [5] K.J. Waldron, G.J. Kinzel, Kinematics, Dynamics, and Design of Machinery, John Wiley, 1999. [6] H. Uchida, K. Nonami, Quasi force control of mine detection six-legged robot COMET-I using attitude sensor, in: Fourth International Conference On Climbing and Walking Robots CLAWAR 2001, Professional Eng. Publishing, London, 2001, pp. 979–988. [7] L. Martinis-Filho, J. Silvino, F. Lida, QUARP––Towards a prototype of a quadruped walking robot, in: Fifth International Conference On Climbing and Walking Robots CLAWAR 2001, Professional Eng. Publishing, London, 2002, pp. 611–618. [8] The Hermes Robot, User Manual––IS Robotics, 1994. [9] A. Premont, An investigation of the kinematic control of a six-legged walking robot, Mechatronics 4 (8) (1994) 821–829. [10] K. Makoto, A. Minoru, T. Kazuo, A hexapad walking machine with decoupled freedoms, IEEE Journal of Robotics and Automation Ra-1 (4) (1985) 183–189. [11] D.J. Manko, A General Model of Legged Locomotion on Natural Terrain, Kluwer Academic Publishers, 1992. [12] S. Hirose, A study of design and control of a quadruped walking vehicle, Robotics Research 3 (2) (1984) 113–133. [13] T. Zielinska, T. Goh, K.C. Choong, Design of autonomous hexapod, in: IEEE Workshop on robot Motion and Control, Technical University of Zielona, Gora, 1999, pp. 65–69. [14] J.J. Craig, Introduction to Robotics, Mechanics and Control, Addison-Wesley, 1986.
478
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
[15] Z. Debao, K.H. Low, T. Zielinska, An efficient foot-force distribution algorithm for quadruped walking robots, Robotica 18 (2000) 403–413. [16] T. Zielinska, J. Heng, T. Goh, Design of a mechanical and control system of a walking machines, Journal of Theoretical and Applied Mechanics 3 (38) (2000) 693–708. [17] T. Zielinska, J. Heng, Development of a walking machine: mechanical design and control problems, Mechatronics 12 (2002) 737–754.
Mechanical design of multifunctional quadruped T. Zielinska
a,b,*
, John Heng
a
a
b
NTU––Nanyang Technological University, Singapore Institute of Aircraft Engineering and Applied Mechanics, Warsaw University of Technology, UI. Nowowiejska 244, Warsaw 00 665, Poland
Received 10 July 2001; received in revised form 24 October 2002; accepted 8 November 2002
Abstract This paper describes the mechanical design of a multifunctional four-legged walking machine that is being developed at the Robotics Research Centre, NTU. The major factors influencing the design requirements include minimisation of the weight of the machine, large workspace of the legs, good energy efficiency and relatively high walking speed. The designed walking machine can adopt a variety of configurations such as insect, mammalian, reptile, or human like. The design is invertable and the machine using the legs as manipulators can even perform basic pick and place functions. Lengths of machine leg segments are: 0.12 m––thigh segment and 0.14 m––shank segment, body has a square shape with the side 0.08 m. Total weight is 1.5–2 kG (depends on the type of feet attached and construction of the body frame). The speed of motion depends on the type of implemented gait (side walking with separated legs motion and body transfer, turning motion, stair climbing etc.), during basic straight line quadruped crawl (only one leg at a time in transfer) on a horizontal surface it is 0.045 m/s. It should be stressed that the speed depends on leg configuration and the step length. For shorter steps the number of leg transfers for a given travelling distance is larger and the walking speed is lower. On the other hand, a shorter step length implies that the leg resultant mass will be transferred over a shorter distance in relation to the body frame and the total displacement of the machine centre of mass will be smaller which increases the vehicle stability compared with the situation when the step is longer. Considering different gaits with diverse configurations we can summarise that the walking speed range of the legged autonomous vehicle machine is between 0.001 and 0.045 m/s. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Walking machine; Legged machine; Invertable quadruped; Quadruped design
*
Corresponding author. Address: Institute of Aircraft Engineering and Applied Mechanics, Warsaw University of Technology, UI. Nowowiejska 244, Warsaw 00 665, Poland. E-mail addresses: [email protected] (T. Zielinska), [email protected] (J. Heng). 0094-114X/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00004-1
464
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
1. Introduction There have been many publications underlying that legged locomotion is the best form of locomotion through varying terrain, as compared to using wheels. The surface of a terrain may be uneven, soft, muddy and generally unstructured. Legs in the biological environment demonstrate significant advantage in such situations. It is on this basis that the motive to design and develop a highly flexible walking machine to mimic our four-legged biological cousins begins. Designers of legged machines must take several decisions, which influences the technical features of these systems. As the most important of these, we can list: design of the mechanical structure and choice of leg configuration (choice of number of legs, their kinematics structure, joint design solution), specification of actuating and drive mechanisms (choice of motors type, evaluation of their power, design of motor placement and evaluation of methods of motion transmission from motors to the legs joints), evaluation of expected power consumption in relation to the machineÕs weight, payload, motion conditions (soft, hard terrain, inclined terrain, etc.) and assumed method of walk (speed of motion, number of legs supporting the body during walk, etc.). An important consideration is the adequate design of the control system (on board/ of board control system, control software, hardware and software control systems architecture, distribution of the on board utilities, cables, sensors which influences the stability conditions, etc.). Machines autonomy depends on the internal sensors delivering the information about the internal state of the device and on the external sensors detecting the external environmental conditions. This information must be properly used by the control software that finally determines the machineÕs ÔintelligenceÕ. Our paper deals with the problems of mechanical design which determines the walking machine posture, energy efficiency, range of walking speed and types of gaits which can be later implemented.
2. Overview of the most common legs design ideas Several basic configurations exist for walking machines that have been developed so far. Fig. 1 shows the schematic diagram of an insect leg type. This pattern is used in leg design of average size hexapods––[1–3]. (i.e. LAURON I: 12–15 kG weight, 0.6 m body length, 0.5 m body width, and Hamlet, Katharina, Masha with comparable parameters). Fig. 2 shows the pantograph design type. The points at A and B move linearly (usually driven by a lead screw motor) and if the mechanical linkage is designed carefully, point F can be made to closely follow a straight line (i.e. heavy ASV [4,5], Autopod 0.33, MECANT, Plustech, RIMMO, and average size lighter hexapod COMET [6], quadrupeds QUARUP [7] and TITAN [1]). Fig. 3 shows a leg configuration that is commonly used in a variety of light, small legged vehicles (i.e. [1,8,9]). This design employs two servo motors attached back to back and each motor is responsible for either the lift or swing axes. This design if employed carefully allows the design to be invertable [9].
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
465
Fig. 1. Leg structure typical for insects.
Fig. 4 shows a hybrid leg design which couples the pantograph technique with a linear actuator at its end to generate leg lift and leg swing motions (i.e. MELWALK, MELWALK MARK III [10]). Fig. 5 shows a 3 DOF leg design which employs linear actuation for the leg lift and linear (stretch) motions which employ a rotary actuator for the leg swing actions (i.e. AMBLER [11], DANTE, ROWER [1]). According to our knowledge, in current designs, individual motors are responsible for individual (or single) actuation of the leg joints.
Fig. 2. Pantograph type leg.
466
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 3. Typical two degree of freedom leg.
Fig. 4. Hybrid design.
Fig. 5. Leg with two prismatic joints.
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
467
3. Design considerations 3.1. Mechanical design It was proposed that a collective motor actuation design should be employed. This idea utilizes the collective effort of 2 motors to cooperatively control leg lift and swing action. Fig. 6 shows the current development state of the legged autonomous vehicle (LAVA) developed at the Robotics Research Centre, NTU, Singapore. To minimize the weight and increase energy output of each servo motor, each leg with its 3 DOF is driven through an inverse differential gear drive. Two motors are located in the hip section and the third is in the knee section. The hip motors work collectively to generate the combined leg swing and lift. The benefit of such a system is that the leg lift motor does not have to carry the leg swing motor as if it was a dead weight in conventional leg actuation design. Each of the servo motors is coupled through a worm gear system to ensure a stable system lock to allow the motors to power down when the legged vehicle merely needs to stand thus saving electrical power. Each individual leg section was carefully designed to provide individual lift and swing angles beyond 180° thus providing the vehicle with a high degree of motion dexterity to perform a variety of configurations beyond the basic walking and support functions. Special care was taken to ensure that minimal obstruction was encountered between the legs and the body when the leg were being driven through its large leg lift and sweep angles. Furthermore, minimal modifications would be required for LAVA to be waterproofed. A close up drawing of each individual leg is shown in Fig. 7. Each of the hip motors is rated at 4.5 W (MAXON motor) coupled through a 2 stage 19:1 planetary gear box and a 40:1 worm gear. The knee motor is rated at 3 W with a 3 stage 76:1 planetary gearbox and a 40:1 worm gear. All motors are coupled to a 16 counts per turn 2 phase TTL compatible magnet digital encoder. At an average motor input speed of 3850 rpm, the thigh swing and lift speeds are calculated to achieve 30°/s while achieving a combined torque (both motors working collectively) of up to 4 Nm. This figure includes the efficiency and friction losses at the worm and gearboxes. According to the manufacturer, the epicyclic gearbox has an 81% efficiency and the worm gear is estimated to have a 70% efficiency. The use of a differential gearbox for the thigh motors warrants a high degree of mechanical precision and tight tolerance (0.1 mm) to ensure a smooth and fluid action of the thigh motion
Fig. 6. The current development state of the LAVA.
468
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 7. Close up drawing of individual leg with differential drive.
Fig. 8. Different LAVA configuration; concept (a) actual implementation (b).
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
469
Fig. 8 (continued)
with minimal backlash and friction through it wide swing and lift action. Careful bearing selection, design and orientation were required to ensure that seizure, mechanical wear and heat build up was minimized.
470
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 9. LAVA Ôback to bellyÕ transfer.
3.2. Multifunctional quadruped capability The following figures show LAVA employed in a variety of configurations. In Fig. 8 the drawings on the left (a) shows the intended conceptual LAVA configuration and the corresponding photograph on the right (b) shows the actual implementation. LAVA has overturning capabilities which are useful for the exploration. Scientific instruments like probes, sensors etc. cannot be located symmetrically on the body of the machine as they usually are complex and expensive. If during a mission, a robot topples over and falls down with its back on the ground, it will be able to get up and return to the normal position. Fig. 9 illustrates the sequence of postures designed for LAVA Ôback to bellyÕ active transition [14], and Fig. 10 shows LAVA in one of those postures.
4. Discussion of advantage and disadvantages of proposed design 4.1. Differential mechanism and energy/torque demand LAVA leg actuation employs a differential gear drive and worm gears which offer a self-locking feature. So far as to the authorsÕ knowledge, this combined solution has not been implemented in
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
471
Fig. 10. Lava between posture 9 and 10.
any existing known walking machines. Mechanical locking of the leg in support phase has been proposed by Hirose [12] but many of the existing legged machines offer a back drivable actuation system. The self-locking feature contributed significantly in aiding the torque required during the leg support phase. Our goal was not only to attain good mechanical efficiency but also large leg work-space which offers the ability to walk with many different gaits and with different legs postures. Moreover, LAVA has an active over-turn ability. This property is exhibited only by the hexapod IOAN with tip-over by active motion. Such feature is useful when the device is used for exploration. Scientific instruments like probes, sensors etc. cannot be located symmetrically on the body of the vehicle as they are expensive. Fig. 11 shows the expected torque demand in quadruped crawl gait with Fig. 12 showing the angles related not to the joints but to the motors (angular motions in hip joints were recalculated
Fig. 11. ALV2 (2 kG) torque demand during quadruped crawl.
472
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 12. Motors motion during leg transfer phase. Angles are related to the expected angle contributed by motor rotation (h1 þ h2 for one hip motor, h1 h2 for second hip motor).
to the angles which were contributed by the motors through the differential mechanism). For simplicity in both drawings, the horizontal axis is expressed in scaled units of control time. In Fig. 11, one control step takes in reality of 60 ms. Hence the support phase takes 15 60 ms ¼ 900 ms. This duration of the support phase has been used in the majority of implemented gaits. In Fig. 12 one control step is 20 ms, so the transfer phase takes 300 ms. The torque was calculated using the Jacobian approach, the real values of reaction forces were considered as the input values (for horizontal terrain). In calculations, the torque required for supporting the body and for powering of the body motion were distinguished. The largest torque required (Fig. 11––line with circles) was for the support of the body. With an ideal self-locking mechanism in place, this torque demand can be neglected and (in hip joint) we need to consider only the torque consumption for machine motion. Assuming self-locking mechanism to be highly effective, we can conclude that if the motors are chosen to deliver driving torque, this torque instead of supporting the machine (and load) itself can be used to produce the machine motion with extra payload. In other words this torque can be used to compensate for a bigger load which demands bigger body motion powering torque. In classic legged vehicle design this is not achievable. In actual practice, the self-locking mechanism does not approach 100% effectiveness and moreover the body powering torque significantly increases when walking on uneven terrain or when walking by turning/side motion. Non-back drivable nature of the worm gear is evident in standing support, but it helps also to support the body during walking. In this situation the load distributed over the legs is not able to force the Ônon-usableÕ rotation of the worm gears and the motor shafts connected to them. We tested a family of GROVEN (GROVEN I, II, III) hexapods (35–45 kG) with legs designed similarly to the legs of LAVA [13]. In the case of the
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
473
first GROVEN prototype (45 kG) the design procedure neglected the self-locking capability of the worm gears and assumed 0 kG payload. In consequence the real (measured) payload of the machine was 40 kG in tripod gait. During estimation of the motor parameters, it is suggested that the demand for torque supporting the weight of the body cannot be fully ignored. We need to add a safety margin to cover the different walking conditions (despite of the margin which must be added to cover the limited mechanical efficiency and presence of friction). In our works, on the evaluation of motors for the next GROVEN prototypes, we did not add the torque covering the limited efficiency but instead we neglected the torque benefits resulting from the self-locking. The experiments with LAVAÕs different gaits over different surfaces (with the inclination up to 15°) showed that the motors and reduction ratios were selected properly. 4.2. Differential mechanism and walking machine speed demand For the discussion provided below we assumed that the reductions are equal in classic design and design with differential gear. In differential gear mechanisms, one motor moves following the sum of two angles at the hip joint (angle related to leg motion up/down plus angle related to the backward/forward leg-end motion). Another hip motor follows the difference of hip angles. As a result one of the hip motors (depends on anglesÕ sign) must move with a velocity higher than that expected in classic hip design. We noticed that in the implemented gait, one of the LAVA hip motors is expected to cover the angle change 2.7 times bigger (Fig. 12) than the angle increment in classical design where separate motors powers separately forward/backward and upward/downward thigh link motion. It means that the speed expected from motor in differential mechanism can be 2–3 times bigger than in classic design. For the same gear reduction classically designed machine will be about twice the speed of LAVA. Other factors which are limiting the body traveling speed is the number of leg steps which must be completed to cover the given body distance. If the leg step length (the range of leg-end backward shift in relation to the body in one support phase) is shorter, the number of steps over the distance increases thus increasing the time spent on ineffective transfer phase (raising up and moving down the leg). Hence the step length should be long. However a large increase in the step length over the evaluated limit will cause instability especially when walking on undulating or soft terrain.
5. Optimization of leg proportions considering the power demand The length of leg links was chosen after the study of total power demand (power demand for 3 motors) for one leg during the support phase. Seven listed further configurations were considered. For each configuration, the energy consumption in relation to the thigh/shank length proportion was calculated. For the final LAVA design, the leg links proportion was chosen such that this proportion was shown to have relatively low power demand for different leg configurations.
474
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
First the torque in the leg joints were calculated, using (as mentioned) the Jacobian approach [14]: s ¼ J ðhÞT F
ð1Þ
where J ðhÞT is the transposed Jacobian matrix [m]. F represents the force vector exerted by the leg to support and move the body. The components of the force vector are equal to reaction forces, but with opposite signs F ¼ ½fx ; fy ; fz T ½N
ð2Þ
Jacobian J ðhÞ is expressed in the same frame as the force F (base frame––x0 ; y0 ; z0 ), and s is the vector of joint load torques [Nm]. The direct and inverse kinematics problem was solved for the legs [9]. The typical Denavit–Hartenberg approach was applied. For the inverse problem the algebraic approach was applied. As this problem is not complex and kinematics of 3 DOF mechanism is very often described in the robotics books we will not go into details. The leg-end forces were provided for the solution of the so called force distribution problem which is well known for scientists working on walking machines (i.e. [15]). The expected hip motors power Phip was approximated by Phip ¼ ðs1 þ s2 ÞDh=ðts 360°Þ
ð3Þ
The reduction ratio minimizing the motor torque requirement was chosen paying the attention to walking speed. If this speed is equal to v, and step length is equal to s, the time ts of support phase is v=s. The biggest angular velocity is expected during leg transfer phase. This value strongly depends on the gait pattern and shape of the leg-end trajectory during transfer phase. In quadruped crawl gait with duty factor 0.75, the leg supports the ground by 75% of the gait period and only 25% of this period leg is in the transfer phase, which means (approximating) that angular speed in transfer must be at least 3 times bigger than that in support. We assumed that in the worst case, the transfer can be 4 times shorter than the support. This can also cover a bigger range of angle change in transfer due to the different shape of leg-end trajectory than that in support. From there, the joint angular speed is equal to h_hip ¼ 4Dhmax
hip =ts
ð4Þ
where (differential mechanism): Dhmax
hip
¼ maxfjDh1 þ Dh2 j; jDh1 Dh2 jg
The reduction evaluated for this speed is equal to the average motor speed divided by the joint angular speed. Average motor speed mv is expressed in rpm (rotation per minute), assuming that unit of time ts is also minute, the reduction ri is equal to rhip ¼ mv =ð4Dhmax
hip =ð360° ts ÞÞ
ð5Þ
Neglecting power (energy) benefit resulting from the self-locking, but not considering the limited mechanical efficiency, we can express that the expected hip motors torque is equal to
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
sm1 þ sm2 ¼ ðs1 þ s2 Þ=rhip
475
ð6Þ
Considering in formula (3) Eqs. (5) and (6) hip motors power Phip can be approximated as Phip ¼ ðsm1 þ sm2 Þmv
ð7Þ
In power demand analysis we used formula (1) for torque evaluation, next the gear ratio using (4) and (5) was calculated and from (6) and (7) motors power were evaluated. The knee motor power Pknee was calculated in similar way, the analysis of power demand takes into account the total torque P ¼ Phip þ Pknee . In the study of power demand the following configurations were considered––Fig. 11 (in a list below, initial value on the beginning of support phase is given) Configuration 1: Configuration 2: Configuration 3: Configuration 4: Configuration 5: Configuration 6: Configuration 7:
h1 ¼ 42°, h2 ¼ 45°, h3 ¼ 98° (insect configuration) h1 ¼ 42°, h2 ¼ 0°, h3 ¼ 90° (typical reptile configuration) h1 ¼ 42°, h2 ¼ 10°, h3 ¼ 80° (reptile configuration) h1 ¼ 42°, h2 ¼ 10°, h3 ¼ 92° (intermediate configuration: insect–reptile type) h1 ¼ 42°, h2 ¼ 20°, h3 ¼ 94° (insect configuration) h1 ¼ 42°, h2 ¼ 30°, h3 ¼ 50° (intermediate configuration: reptile–mammal type) h1 ¼ 42°, h2 ¼ 30°, h3 ¼ 96° (insect configuration)
As shown in Fig. 14, in many cases (i.e. configurations) a ratio with l1 =l2 lying within the range of 0.3–0.9 is associated with smaller motor power rather than that for other leg links proportions. For all the cases considered, insect Configuration 7 is optimal (low motor power requirements) for l1 =l2 greater than 0.6. Such a low power requirement can be observed in other configurations only for exactly fixed l1 =l2 (e.g. for Configuration 5, where l1 =l2 ¼ 0:3 or for Configuration 4 where l1 =l2 ¼ 0:48).
Fig. 13. General view of leg (insect configuration).
476
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
Fig. 14. Influence of leg configuration and proportion on motor power.
Despite of the lower power demand for l1 =l2 greater than 0.5, we can notice that for l1 =l2 > 0:5 the power requirement for one configuration can be doubled when compared with another one. For example, for l1 =l2 equal to 1, the power requirement depends noticeably on the configuration and is in the range of 2 W (Configuration 7) to 3.8 W (Configuration 1). On the other hand, for the ratio smaller than 0.3, the range of power change in relation to configuration is much greater, e.g. 2.25–7.25 W for l1 =l2 ¼ 0:2. Summarising––for the designer––the ratio l1 =l2 between 0.6 and 0.9 (to 1.0) can be suggested as the optimum choice as for that ratio, in all legs configuration motor power is minimal or is not far from minimal demand. In the design of LAVA legs l1 =l2 ¼ 0:857 was chosen (see Fig. 13).
6. Conclusions The mechanical structure of the prototype must be very well assembled for effective selflocking. When this is the case, smaller torque in the hip is required and gear reduction can be decreased which increases walking speed. As the payload and speed are inversely proportional in relation to the gear reduction, differential gear has the advantage in torque when compared to the classic design (torque benefit depends on the quality of manufacturing). The expected gain in payload can even be doubled while speed reduction is less than 2 times (for equal motors and reductions)––which means that by decreasing the reduction gear by two in our prototype we can achieve similar payload and speed when compared with classic machines (with two time greater reduction).
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
477
For the better energy efficiency the leg link proportion and body size influencing the walking step length must be designed properly. The proposed novel leg design can be utilized in every multi-legged statically stable machine. The four legged machine LAVA has been built according to our design considerations. Multi-level real time based control system was developed and the prototype moves successfully using several types of gaits. [16] presents a detailed discussion of motor power evaluation considering the assumed vehicle mass and expected motion speed, [17] describes the LAVA control system and experiments with compliance position force control. The development and usability of walking machines can be constantly improved by the aid of proper design. The final goal of this presented work is to develop an autonomous walking machine, which will be optimised from the point of view of energy consumption and will be able to operate autonomously in natural conditions.
Acknowledgement The authors would like to express their thanks to Dr. G. Seet, Director of the Robotics Research Centre, NTU who supported the conducted research.
References [1] K. Berns, Walking Machines Catalogue (www.fzi.de/ids/WMC). [2] W. Ilg, K. Berns, A learning architecture based on reinforcement learning for adaptive control of walking machine LAURON, Robotics and Autonomous Systems 15 (1995) 321–334. [3] U. Muller-Wilm, H. Cruse, J. Eltze, J. Dean, H.-J. Weidemann, F. Pfeiffer, Kinematic model of a stick insect as an example of a six-legged walking system, Adaptive Behavior 1 (2) (1992) 1–15. [4] K.J. Waldron et al., Configuration design of the adaptive suspension vehicle, Robotics Research 3 (2) (1984) 37–48. [5] K.J. Waldron, G.J. Kinzel, Kinematics, Dynamics, and Design of Machinery, John Wiley, 1999. [6] H. Uchida, K. Nonami, Quasi force control of mine detection six-legged robot COMET-I using attitude sensor, in: Fourth International Conference On Climbing and Walking Robots CLAWAR 2001, Professional Eng. Publishing, London, 2001, pp. 979–988. [7] L. Martinis-Filho, J. Silvino, F. Lida, QUARP––Towards a prototype of a quadruped walking robot, in: Fifth International Conference On Climbing and Walking Robots CLAWAR 2001, Professional Eng. Publishing, London, 2002, pp. 611–618. [8] The Hermes Robot, User Manual––IS Robotics, 1994. [9] A. Premont, An investigation of the kinematic control of a six-legged walking robot, Mechatronics 4 (8) (1994) 821–829. [10] K. Makoto, A. Minoru, T. Kazuo, A hexapad walking machine with decoupled freedoms, IEEE Journal of Robotics and Automation Ra-1 (4) (1985) 183–189. [11] D.J. Manko, A General Model of Legged Locomotion on Natural Terrain, Kluwer Academic Publishers, 1992. [12] S. Hirose, A study of design and control of a quadruped walking vehicle, Robotics Research 3 (2) (1984) 113–133. [13] T. Zielinska, T. Goh, K.C. Choong, Design of autonomous hexapod, in: IEEE Workshop on robot Motion and Control, Technical University of Zielona, Gora, 1999, pp. 65–69. [14] J.J. Craig, Introduction to Robotics, Mechanics and Control, Addison-Wesley, 1986.
478
T. Zielinska, J. Heng / Mechanism and Machine Theory 38 (2003) 463–478
[15] Z. Debao, K.H. Low, T. Zielinska, An efficient foot-force distribution algorithm for quadruped walking robots, Robotica 18 (2000) 403–413. [16] T. Zielinska, J. Heng, T. Goh, Design of a mechanical and control system of a walking machines, Journal of Theoretical and Applied Mechanics 3 (38) (2000) 693–708. [17] T. Zielinska, J. Heng, Development of a walking machine: mechanical design and control problems, Mechatronics 12 (2002) 737–754.