Robust Control of a Walking Robot System and Controller Design

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 174 (2017) 947 – 955

13th Global Congress on Manufacturing and Management, GCMM 2016

Robust Control of a Walking Robot System and Controller Design Kunquan Li, Rui Wen* Henan Institute of Engineering, No. 1, Xianghe Road, Xinzheng, Zhengzhou 451191, China

Abstract In this paper, the method of multi-body dynamics is employed to construct the equations of motion for the complicated biped robot system and designed a robust controller for stable dynamic biped robot system. Using this approach, the planned control law can be adjusted continuously in response to the current movement of the feet of the robot, and this adjustment can be accomplished in a very smooth way. Under the assumption of the bounded estimation errors on the unknown parameters, the proposed controller provides a successful way to achieve the stability and can provide a powerful safeguard for the robot system. Simulation results are presented to show the system’s efficiency and stability in adapting to an uncertain terrain. © 2017 2016The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management Keywords: robust controller; walking robot; multi-body dynamics; bounded estimation errors;

1. Introduction One of the main reasons to develop a four-legged walking robot is to overcome the lack of mobility of wheeled vehicles on irregular terrains. The ability to traverse uneven or varying terrain at high speeds, turn sharply, and start or stop suddenly are all ordinary aspects of four-legged locomotion for a variety of cursorial mammals. Researches on four-legged robots have been widely carried out [1-3]. A quadruped robot with Multi-degrees of freedoms is a high nonlinear system. Firstly, sometimes the uncertain parameters and the external disturbance apply a very adverse effect on the control performance. Secondly, motion of a quadruped robot under the four legs full support can be regarded as a dynamic system under holonomic constraint, and it make the control more complicated. Robust control maybe a suitable method to solve the problems, but the literatures that use robust control to control the quadruped robot is sparse. E.H. Mamdani applied fuzzy control to

* Corresponding author. Tel.: +86-0371-62508765; fax: +86-0371-62508765. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 13th Global Congress on Manufacturing and Management

doi:10.1016/j.proeng.2017.01.246

948

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

practical robot initially [4-6]. Then, fuzzy system has been widely used and studied in robot modeling, flexible arm control, position control, fuzzy compensation control and tracking design of robot system [7,8]. In addition, variable structure control system had developed some improved algorithm to weak chattering, such as dynamic adjust sliding mode parameter and line estimate sliding mode parameter [9-11]. In this paper, we present a method for designing free gaits for structural symmetrical four-legged robot capable of performing statically stable, omnidirectional walking on irregular terrain. Robot’s virtual model is constructed and a control algorithm is proposed by applying virtual components at some strategic locations. For adjusting the planned control law continuously in response to the current movement of the feet of the robot, a control scheme has been presented to make the quadruped robot control systems robust to parametric and unstructured uncertainties. 2. Description of the walking robot system 2.1. Walking machine and foot contact model The four-legged robot model described in this section is shown in Fig. 1 and the nominal parameters of leg 1 are shown in Fig. 2 and Fig. 3. Unlike some other studies, the model has mass in each of the leg links and a symmetric distribution of mass in the trunk, and the body’s center of mass is not located at its geometric center. Articulated knee joints are used instead of prismatic joints to better model biological legs. The virtual leg is measured from the body’s normal to the imaginary line connecting the hip to the foot, while the virtual leg length is the hip-to-foot distance. The virtual leg convention described here effectively transforms an articulated leg to a prismatic leg, which is more convenient for control purposes.

Fig. 1. Four-legged robot model.

Fig. 3. Parameters of swing leg.

Fig. 2. Parameters of stand leg and presented virtual leg.

Fig. 4. The simplified, virtual leg model for computing the approximate induced foot force.

949

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

The no-slip condition is computed by using static and kinetic friction coefficients to model the interaction of rubber-coated feet on concrete. Joint torques, W L , W LD and W LE are bounded such that the approximated resultant foot force remains within the friction pyramid to reduce slipping. A pyramid is employed instead of the standard friction cone to simplify the computation by decoupling each component of the foot force. This avoids having to solve a coupled equation and dramatically simplifies the calculations. The algorithm uses a simplified approach where a virtual leg (Figure 4) is used to approximate the induced contact force acting on the ground due to joint torques , W L , W LD and W LE of TL , TLD and TLE . Ideal actuators are modeled at each of the abductor/adductor, joints so that the results of this study are independent of assumptions regarding specific actuator models. Dynamic simulation is implemented using a software package developed for tree-structured robots, and Runge-Kutta fourth-order integration with a 0.5 ms time step. Yaw, pitch, and roll are defined using the Z-Y-X Euler angle convention described in[12,13]. 2.2. Dynamics formulation Let T  5 denote the joint coordinate of the four-legged robot, where, n is the rigid links of robot. When the robot links is rigidly in contact with an uncertain surface, the environmental constraint is expressed as an algebraic  , where )  5 Q u 5 6 5 P . Denote equation of the coordinate T and time W , namely, )T W $)T W as the Jacobian matrix of )T W with respect to T , i.e., $ w)T W wT . Without loss of generality, the uncertain constraint is decomposed into a nominal part ) T W and a constraint modeling error part ) T W . This implies that the Jacobian matrix $T W $ T  '$T W with nominal $ T and uncertain '$T W . Meanwhile, several assumptions on the system are made as follows. Assumption 1 The Jacobian matrix $ is bounded for all T and W . Assumption 2 The robot manipulator is operated away from any singularity. In this case, the Jacobian $ is of 7 full row rank m, such that the joint coordinate T is partitioned into T T7 T7 for T  5 Q  P and T  5 P , where T :TW with a nonlinear mapping function :˜ from an open set 4  5 Q  P u 5 P to 5 .   Assumption 3 The terms w: wT , w : wT , w: wW and c are bounded in the workspace. The dynamic equation of the constrained robot is written as Q

>

M (q)q  C (q, q )  F (q, q, t ) W  W d

Y1 (q, q, q)K

@

(1)

Where, 0 T , &T TT , W are the inertia matrix, coriolis/centripetal force, and joint driving torque, respectively; )T TW denotes a combinational term of gravitational force, joint stick-slip friction force, and external joint/environment disturbances; W G is the vector of any generalized input due to disturbances or unmodeled dynamics; and K is a constant m-dimensional vector of inertia parameters and <  is an Q u P matrix of known functions of T , T and T . 3. Description of the walking robot system It is assumed that there exists a known bound

K~

K  K0 d U

U

on parametric uncertainty such that

(2)

Where, K represents the fixed parameters in dynamic model. Also, for unstructured uncertainty it is assumed that the norm of W G W is bounded as

W d  w1  w2 q  w3 q

(3)

950

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

Where, Z  , Z  and Z  are known positive constant[14]. Equation (3) can be written as

Wd

W TQ

(4)

>

>

@

@

7

Where, :  T  T . Z  Z  Z  and 4 The robust controller design problem is as follows: given the desired joint trajectory TG W , derive a control law for joint torques such that the manipulator joint position TW accurately tracks TG W in the presence of parametric and unstructured uncertainties. Joint position and velocity errors are defined as H TG  T and H TG  T , respectively. Define

r

e  /e; Q

qd  /e

(5)

Where, / is a constant diagonal positive definite (pd) matrix. Assume that T G is a given twice continuously differentiable reference trajectory. As mentioned before, the robot dynamics is linear in parameter, thus we can write

M (q)Q  C(q, q )Q  F (q, q, t ) Y1 (q, q,Q ,Q)K

(6)

Substituting equations (5) and (6) into equation (5), yields

M (q)r  C (q, q )r

Y1 (q, q,Q ,Q)K  W  W d

>

7

(7)

@

7 7

Equations (2) and (4) indicate that [ H U can be considered as the state vector of the system. Now, we consider three compensation parts for the control law. In the first part, we define a nominal control law as

W0

Y (q, q,Q ,Q)K0  K r

(8) Where, K represents the fixed parameters in dynamic model and . U is the vector of PD control effort. The second and the third parts are considered to compensate for Ka and W G , and are denoted by X U and X G , respectively. Thus the control law is

W

Y (q, q,Q ,Q)K0  Kr  uU  ud

We define

XU

and

(9)

XG as

uU

T T ­ °UYY r Y r ® ° ¯0

ud

­W T Qr r ® ¯0

YTr z 0 YTr rz0 r

0

(10)

0

(11)

951

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

To analysis the stability of the whole system the following Lyapunov function candidate is defined.

V ( x, t )

r T Mr 2  eT /Ke

(12)

Where, . is a constant diagonal pd matrix. Note that T TG  H , as a result 0 T 0 [ W . Computing the time derivative of 9 along the trajectory and making some simplification by considering equations (1) and (5), yield

V 2eT /Ke  r T (Mq  Cq  F  W d  W  M/e  M r 2)

(13)

T Q  U and Q TG  /H . Also, due to skew symmetric property U 7 0T  &T T U  U  5 . Therefore, Equation (13) is simplified to

>

Based on (5),

V

@

2eT /Ke  r T (MQ  CQ  F  W d  W )

(14) Which based on equation (6) is arranged as

V

2eT /Ke  r T (YK  W d  W )

(15)

Substituting equation (7) into equation (15) and simplifying the result, we have

V

V1  r T (YK~  uU  ud  W d )

Where, 9

V

(16)

H7 /./H  H7 .H . Equation (16) can be written as the inequality

V1  r T Y K~  r T uU  r T W d  r T ud

(17)

It follows from equations (2), (4) and (17) that,

V

V1  z

(18)

rY T U  r T uU  r T W T Q  r T ud

(19)

Where,

z

Substituting equations (10) and (11) into equation (19) gives

]

 , thus according to (18) we have

952

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

V ( x, t ) d V1 ( x)

(20)

9 is a pd function and based on equation (16), 9 becomes zero at the origin, thus, 9 is negative definite. We can rewrite 9 as 9 [ 7 %[ where B

0 ª/K «0 M (q) ¬

º 2»¼

(21)

0 T is a pd matrix, as a result, %T is a pd matrix. If O%T represents the eigenvalues of %T , and we % PLQ O%T , , % PD[ O%T , , where , is the identity matrix, then

define

T

T

xT Bx d V d xT Bx

(22)

or equivalently

min O ( B(q)) x d V d max O ( B(q)) x 2

q

q

2

(23)

Where O˜ represents the eigenvalues of the corresponding matrix. Equation (22) indicates that 9 is between two class-K functions. Thus, 9 is pd, decrescent and radially unbounded. Based on Lyapunov theorem for nonautonomous systems, the equilibrium point at 0 is globally uniformly asymptotically stable. These results can be summarized in a theorem for the control system. We can define a continuous control law by defining X U and d as

XG

uU

­ UYY T r Y T r ° ® T ° ¯ UYY r H

ud

T ­ °W Qr r ® T ° ¯W Qr G

YTr ! H YTr d H

(24)

r !G r dG

(25)

Where H and G are positive constants. It is obvious that the control law is continuous for any H !  and U ! G then based on equations (18), (24) and (25), we achieve again equation G !  . If < 7 U ! H and 7 (20), thus, 9   . If < U ! H and U d G then based on equations (18), (19), (24) and (25), we have 9 d 9  : 7 4  . If 7 < 7 U d H and U ! G then similar to the second case, we can achieve 9 d 9  UH  . If < U d H and U d G then with an approach similar to previous cases, we have

V d V1  W T Q 4  UH 4

>

@

(26)

The fourth case is the worst case and the inequality equation (26) is satisfied for all cases. On the other hand, if 7 H7 H7 , then 9 [Ц7 3[Ц where we define [Ц

953

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

P

ª/K/ « 0 ¬

It is clear that

3

0º / »¼

(27)

is a pd matrix. 9 is bounded with a class-K function as

2 V d min O ( P) xˆ

(28)

Substituting equation (28) into equation (26) and doing some straightforward calculation, we can show that if

>

xˆ ! (GW T Q  UH ) 4 min O ( P) Then 9

@

0.5

(29)

 .

4. Simulation The results of simulation with quadruped walking robot model discussed in section 2 are presented in this section. The torque during steady-state walking for the leg joints TL are shown in Figure 5.

Fig. 5. Torques for the leg axes.

Fig. 6. Torques for the hip axes.

The asymmetry among the leg joint torque curves occur due to the asymmetric footfall sequence. The leg torques are small relative to the hip and knee joints, since the leg joints rotate by small amounts during flight and have small moment arms relative to the dominant vertical forces during contact. Square waveforms indicate leg contact. The hip torques are shown in Fig. 6. As there is signi-fycantly larger peak torques required during stance. Positive torques during the front leg stance phases result from the shoulder-braking behaviour. The shoulder torque is generally positive during stance to retard the retraction of the front legs. The knee torques are shown in Fig. 7. Minimal torque is required to transfer the shank during flight because only the relatively small inertial load of the shank. Note that an ideal series elastic configuration is assumed.

954

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

Fig. 7. Torques for the knee axes.

5. Conclusions In this paper, a control scheme has been presented to make the quadruped robot control systems robust to parametric and unstructured uncertainties. This controller is designed based on a priori knowledge of uncertainties bounds. A constant bound is considered for parametric uncertainty and a position-velocity dependent bound is considered for unstructured uncertainty. The stability properties of the system with two different control laws were studied and stated via two theorems. For the first control law, it is guaranteed that the whole system is globally uniformly asymptotically stable. The second control law is continuous, and uniform ultimate boundedness of the tracking error is established for it. Simulation results illustrate that the proposed control scheme increases the robustness of the system to uncertainties. Acknowledgment This research is sponsored by the NSFC (Grant No.50675016). References [1] J.H. Shin, K.B. Park, S.W. Kim and J.J. Lee, Robust Adaptive Control for Robot Manipulators Using Regressor-Based Form, IEEE. (1994) 2063-2068. [2] S. Hirose, H. Kikuchi and Y. Umetani, Standard circular gait of a quadruped walking vehicle, Advanced Robotics. 2(1986) 143-164. [3] D. L. Jindrich, R. J. Full, Many-legged maneuver-ability: Dynamics of turning in hexapods, The Journal of Experimental Biology. 202 (2005)1603-1623. [4] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, 1991. [5] Wang F Y and Lever P L A, Rule generation and modification for intelligent control using fuzzy logic and neral networks, work report: The University of Arizona, Tucson, Arizona, USA, 1995. [6] H King L Litz, Inconsistency detection-a powerful means for the design of MIMO fuzzy controllers, Int. Conf. Fuzzy systems. (2006) 1191-1197. [7] Y Stepanenko CY Su, Varible structure control of robot manipulators with nonlinear sliding manifolds, Int J. Control. 2(1993)285-300. [8] WT Miller, Real-time application of neural networks for sensor-based control of robots with vision, IEEE Trans. System, Man, and Cybernetics. (1989)825-831. [9] S. McMillan, D. E. Orin, R. B. McGhee, An object oriented software package for efficient dynamic simulation of underwater robotic vehicles, In Underwater Robotic Vehicles: Design and Control, J. Yuh, Ed. Albuquerque, NM: TSI Press. (2005)73-98.

Kunquan Li and Rui Wen / Procedia Engineering 174 (2017) 947 – 955

[10] T TakagiM Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. System, Man, and Cybernetics. 1(1985)116-132. [11] M Fei, B Chen, Intelligent control method intercross synthesis and application. Control Theory and Applications. 3(1996)273-281. [12] Ananthraman S Garg D P, Training backpropagation and CMAC neural networks for control of a SCARA robot, Engng Applied Artif. Intell. 2(2003)105-115. [13] S. Hirose, Y. Fukuda and H. Kikuchi, The gait control system of a quadruped walking vehicle, Advanced Robotics. 4(1986)289-323. [14] DP. Wong, DE. Orin, Control of a quadruped standing jump over irregular terrain obstacles, Autonomous Robots. 1(1995)111-129.

955