Nonlinear H<FONT FACE=Symbol>¥</FONT> control for underactuated manipulators with robustness tests

Abstracts

In this paper, the model-based robotic control problem with disturbance attenuation (or robotic H<FONT FACE=Symbol>¥</FONT> control problem), presented in Chen et all (1994), is extended to underactuated manipulators. The dynamic coupling between the links is used to control all manipulator's free joints. A global explicit solution is found solving a minimax Bellman-Isaacs equation, generated via differential game theory. Experimental results, obtained from UArm II manipulator, considering fully actuated and underactuated configurations, are presented.

Robust control; nonlinear H<FONT FACE=Symbol>¥</FONT> control; underactuated manipulators

Neste artigo, o problema de controle robótico baseado em modelo com atenuação de distúrbios (ou problema de controle H<FONT FACE=Symbol>¥</FONT> robótico), apresentado em Chen et all (1994), é estendido para robôs manipuladores subatuados. O acoplamento dinâmico entre os elos é usado para controlar todas as juntas livres do manipulador. Uma solução explícita global é encontrada resolvendo um problema minimax definido através de uma equação de Bellman-Isaacs gerada pela teoria dos jogos. Resultados experimentais, obtidos com o manipulador UArm II, considerando as configurações totalmente atuada e subatuada, são apresentados.

Controle robusto; controle H<FONT FACE=Symbol>¥</FONT> não linear; manipuladores subatuados

Nonlinear H_¥ control for underactuated manipulators with robustness tests

Adriano A. G. Siqueira; Marco Henrique Terra

Electrical Engineering Department, University of São Paulo - São Carlos, SP, Brazil; [e-mail], [e-mail]

ABSTRACT

In this paper, the model-based robotic control problem with disturbance attenuation (or robotic H_¥ control problem), presented in Chen et all (1994), is extended to underactuated manipulators. The dynamic coupling between the links is used to control all manipulator's free joints. A global explicit solution is found solving a minimax Bellman-Isaacs equation, generated via differential game theory. Experimental results, obtained from UArm II manipulator, considering fully actuated and underactuated configurations, are presented.

Keywords: Robust control, nonlinear H_¥ control, underactuated manipulators.

RESUMO

Neste artigo, o problema de controle robótico baseado em modelo com atenuação de distúrbios (ou problema de controle H_¥ robótico), apresentado em Chen et all (1994), é estendido para robôs manipuladores subatuados. O acoplamento dinâmico entre os elos é usado para controlar todas as juntas livres do manipulador. Uma solução explícita global é encontrada resolvendo um problema minimax definido através de uma equação de Bellman-Isaacs gerada pela teoria dos jogos. Resultados experimentais, obtidos com o manipulador UArm II, considerando as configurações totalmente atuada e subatuada, são apresentados.

Palavras-chave: Controle robusto, controle H_¥ não linear, manipuladores subatuados.

1 INTRODUCTION

Motion control of manipulators has been the objective of a great number of researches (Chen et all, 1994; Chen and Chang, 1997; Johansson, 1990; Postlethwaite and Bartoszewicz, 1998). Underactuated manipulators, with less actuator than degrees of freedom, are also of interest for many researchers (Arai, 1997; Arai and Tachi, 1991; Bergerman, 1996). The controllability for these mechanical systems and a control strategy were first established in Arai and Tachi (1991). First, all passive joints (without actuator) are controlled to theirs set-point, using the dynamic coupling. Then, with the passive joints locked, the active ones (with actuator) are controlled by themselves. In Bergerman (1996), three possibilities of selecting the joints to be controlled at every control phase are derived. One can select only passive joints, passive and active joints or only active joints.

The effort to control the generalized coordinates of the manipulator (fully actuated or underactuated) to follow a desired trajectory can be a hard task if parameters uncertainties and exogenous disturbances are present. Robust control with a nonlinear H_¥ criterion (Chen et all, 1994; Chen and Chang, 1997) and adaptive control strategy (Johansson, 1990) have been proposed to eliminate the effects of these perturbations. In Johansson (1990), the adaptive strategy is based on optimal motion control with minimization of the applied torques.

H_¥ control for nonlinear time-invariant systems has been widely discussed since the past decade (Isidori, 1992; van der Sachft, 1991; van der Schaft, 1992; Ball et all, 1991; Lu and Doyle, 1991). The nonlinear H_¥ control problem means that we need to find an L₂ induced norm between input and output signals limited by a level g. In Lu (1996), these results are extended to time-varying systems with finite-time horizon.

A robotic H_¥ control problem, or a model-based robotic control problem with desired disturbance attenuation, is proposed in Chen et all (1994). A global explicit solution for this problem, formulated as a minimax (leader-following) game, is developed using differential game theory (Basar and Oslder, 1982; Basar and Berhard, 1990). From this theory, one needs to solve a minimax Bellman-Isaacs equation, which after some rearrange it is redefined as a Hamilton-Jacobi equation found in Lu (1996). The result of Chen et all (1994) is a kind of feedback linearization with a nonlinear term introduced in the control acceleration. Some experimental results were obtained in Postlethwaite and Bartoszewicz (1998) using a similar approach.

The formulation presented in Chen et all (1994) is resumed here as a background to the main results of this paper: the extension of the robotic H_¥ control problem for underactuated manipulators and the application of this methodology in the experimental robot manipulator UArm II, a three link planar manipulator with revolute joints, that can be configured as fully actuated or underactuated.

This paper is organized as follows. In Section 2, the robotic H_¥ control problem is formulated based on Chen et all (1994). In Section 3, this problem is extended to the underactuated case. In Section 4, the solution presented in Chen et all (1994), for this H_¥ control problem, is described. In Section 5, experimental results obtained from the UArm II are presented. In Section 6, the conclusions are presented.

This paper was previously presented in the International Workshop on Underwater Robotics for Sea Exploitation and Environmental Monitoring, held on October 2001 in Rio de Janeiro and organized by Professor Liu Hsu of Federal University of Rio de Janeiro.

2 ROBOTIC H_¥ CONTROL PROBLEM

In this section, the H_¥ control problem is formulated for a manipulator where the disturbances are derived from parametric uncertainties and exogenous inputs, following the line defined in Chen et all (1994).

The dynamic equations of a manipulator can be found by the Lagrange theory as

where q Î Âⁿ are the joint positions, M(q) Î Â^nxn is the symmetric positive definite inertia matrix, C(q,) Î Â^nxn is the Coriolis and centripetal matrix, F₀() Î Âⁿ are the frictional torques, G(q) Î Âⁿ are the gravitational torques and t Î Âⁿ are the applied torques. The parametric uncertainties can be introduced by dividing the parameter matrices M(q), C(q,), F(), and G(q) into a nominal and a perturbed part

where M₀(q), C₀(q,), F₀(), and G₀(q) are the nominal matrices and DM(q), DC(q,), DF(), and DG(q) are the parametric uncertainties. Exogenous inputs, w, can also be introduced, and (1) can be rewritten as

with

The state tracking error is defined as

where q^d and ^d ÎÂⁿ are the desired reference trajectory and the corresponding velocity, respectively. The variables q^d, ^d, and ^d (desired acceleration), are assumed to be within the physical and kinematics limits of the control object. The dynamic equation for the state tracking error is given from (2) and (3) as

where

In order to represent this equation in a canonical form, a control input variable u should be defined. Using the following state-space transformation of (Chen et all, 1994; Johansson, 1990)

and selecting the control input as

where T₁₁, T₁₂Î Â^nxn are constant matrices to be determined later and T₁ = [T₁₁T₁₂], the dynamic equation of the state tracking error (4) can be rewritten as

where

The control input (6) is a selective applied torque, since it affects the kinetic energy only. It is not necessary to optimize the gravitation-dependent torques during the motion (Johansson, 1990). The relation between the applied torques and the control input is given by

where

Equation (9) shows that a control acceleration,

^c, with a nonlinear term, is generated by selecting the control input (6).

The H_¥ control strategy aims to attenuate the effects of disturbance, solving the following performance criterion, with a desired attenuation level g

where Q and R are weighting matrices and (0) = 0. This performance criterion is actually the H_¥ optimal disturbance attenuation problem for the model-based robotic control.

Remark (Chen et all, 1994): Formally, subject to the tracking error dynamics (7) a (full information) H_¥-control problem is to find a state feedback law such that

where

and ||.||_L2 denotes the induced L₂ norm.

3 THE UNDERACTUATED CASE

Underactuated robot manipulators are mechanical systems with less actuators than degrees of freedom. For this reason, the control of the passive joints (joint without actuator) is made considering the dynamic coupling between them and the active joints (with actuator). Here, we consider that the passive joints have brakes. The control strategy consists in controlling all the passive joints to reach the desired positions, applying torques in the active ones, and then turn on the brakes. After that, all the active joints control themselves.

Consider a manipulator with n joints, of which n_p are passive and n_a are active joints. It is known (Arai and Tachi, 1991) that no more than n_a joints of the manipulator can be controlled at every instant. Using this fact, we group the n_a joints being controlled in the vector q_c Î Â^na. The remaining joints are grouped in the vector q_r Î Â ^{n - na}. There exist three possibilities of forming the vector q_c (Bergerman, 1996)

1. q_c contains only passive joints: when n_p > n_a and all other passive joints, if any, are kept locked.

2. q_c contains passive and active joints: all other passive joints, if any, are kept locked.

3. q_c contains active joints.

The control strategy is defined as follows: first, select q_c following the possibilities 1 or 2 (according to n_p), until all passive joints have reached the desired position; second, select q_c following the possibility 3 and control the active joints to the desired position.

The dynamic equation (2) can be partitioned as

where t_r are the torques in the remaining joints and t_c are the torques in the controlled joints. For simplicity of notation, the index 0 representing the nominal system is eliminated from the equations.

For the control strategy 1, t_c = 0 because there is no torque in the passive joint. For the control strategy 2, t_c is defined as t_c = [t_ac 0], where t_ac is the torque in the active joints being controlled. From the second line of (11)

we can isolate the controlled joint accelerations

Introducing a desired reference trajectory to the controlled joints (there is no desired reference to the uncontrolled joints), (12) can be rewritten as

In the state-space form, selecting the state vector as

(13) can be defined as

where

Using a similar transformation like (5), the control input u is selected as

and the dynamic equation of the state tracking error (15) is redefined as

where

Based on (16), the control acceleration can be given by

Equation (18) gives the necessary acceleration to the controlled joints follow the desired reference trajectory. The torques in the active joints can be computed using this control acceleration. One can use another form of representing the underactuated system, similar to (11), partitioning (2) as in Bergerman (1996)

where the indexes a and u represent active and unlocked passive joints, respectively, and b(q,) = C(q,) + F() + G(q) + d. Factoring out the vector _r in the second line of (19) and substituting it in the first line, one obtains

If n_p < n_a, the redundant control can also be considered. In this case, the vector of controlled joints contains only the passive joints, q_c = q_u Î , and the vector of remaining joints contains the active joints, q_r = q_a Î . The partitioned equation is defined as follows

Factoring out the vector

_a in the second line, one obtains

where is the pseudo-inverse of the (n_p × n_a) matrix M_ua and z is an arbitrary number. The applied torques in the active joints can be computed as

For the underactuated case, the performance criterion (10) is also used to attenuate disturbances.

4 ROBOTIC H_¥ CONTROL PROBLEM SOLUTION

The solution of the robotic H_¥ control problem (10) can be explicitly found via differential game theory (Basar and Oslder, 1982; Basar and Berhard, 1990) with an appropriated Lyapunov function (Chen et all, 1994). In this section, a resume of the approach presented in (Chen et all (1994)) to solve this problem is presented.

The performance criterion (10) can be rewritten to define the following minimax problem

with (0) = 0. Defining the cost functional

with the Lagrangian

and introducing the Lyapunov function

the performance criterion (10) can be defined as

According to the differential game theory, the solution of this minimax (or leader-follower) problem is found if there exists a continuously differentiable Lyapunov function V(.,.) that satisfies the following minimax Bellman-Isaacs equation

with terminal condition V( (¥), ¥) = 0. Choosing a Lyapunov function of the form

where P(,t) is a positive definite symmetric matrix for all and t, the Bellman-Isaacs equation is then changed to the following Riccati equation

The corresponding optimal control and the worst case disturbance are given, respectively, by

and

Selecting P(,t) properly and using the skew symmetric matrix N( q,) = C₀ ( q,) +(1/2) ₀ ( q,) (Chen et all, 1994), the Riccati equation (4) can be simplified to an algebraic matrix equation. The matrix P(, t) defined by Chen et all (1994) is given by

where K is a positive definite symmetric constant matrix. The simplified algebraic equation is given by

The optimal control and the worst case disturbance can be rewritten, respectively, as

and

The terminal condition is satisfied for this matrix P(.,.) (Chen et all, 1994). Then, to solve the robotic H_¥ problem, we must find matrices K and T₀ which solve the algebraic equation (21). Let the positive definite symmetric matrix Q be factorized as

The solution of (21) is given by

and

with the conditions: K > 0 and R < g²I. The matrix R₁ is defined via Cholesky factorization

Finally, the design algorithm can be outlined as follows

Step 1. Select a desired level of attenuation, g >0.

Step 2. Select the weighting matrix R > 0 such that l_max < g² and the weighting matrix Q as (23), satisfying K >0.

Step 3. Calculate the Cholesky factorization (25) and T₀ (24).

Step 4. Obtain the optimal control u* (22) and the optimal applied torque (8).

Considering the underactuated state tracking error (17), P(,t) can be chosen as

since the matrix M_cc is symmetric positive definite. Note that N_cc ( q,) = C_cc ( q,) + (1/2) _cc ( q,) is also skew symmetric, then the design algorithm used to the totally actuated case can be applied to the underactuated case.

5 EXPERIMENTAL RESULTS

To validate the proposed H_¥ control solution, it is applied in our experimental underactuated manipulator UArm II (Underactuated Arm II), designed and built by H. Ben Brown, Jr. of Pittsburgh, PA, USA (Figure 1). This 3-link manipulator has special-purpose joints containing each one an actuator and a brake, so that they can act as active or passive joints. The manipulator configuration can be changed enabling or not the DC motor of each joint. Optical encoders with quadrature decoding are used to measure the joint positions. Joint velocities are obtained by numerical differentiation and filtering.

For interfacing between computer and manipulator, an input-output interface Servo-To-Go board is used. The board driver is accessed by dynamically linked libraries (dlls) compiled in the MatLab workspace by use of C++ program that contain mex-functions.

A control environment was developed in a suitable way that all changes of configuration and the robot action can be done in a user friendly way. The UMCE (Underactuated Manipulator Control Environment) is written in Matlab language and it is possible to see the real robot motion reproduced in its graphical interface (Figure 2). Simulation tests can also be done in this environment.

All possible configurations, according to active (A) and passive (P) joints location in the arm, are accepted: AAA, AAP, APA, PAA, APP, PAP, and PPA. For example, the configuration AAP means that joints 1 and 2 are active and joint 3 is passive.

The matrices M(q), C(q,), and G(q) of (1) are easily found via Lagrange theory for planar manipulators (Craig, 1989) (See ^{Appendix A}). However, the term F() is determined according to the kind of frictional torques acting in the robot. In this work, we select a velocity-dependent frictional term F() as

where the values f₁, f₂, and f₃ are selected after empirical tests. The manipulator's kinematic and dynamic nominal parameters, which are used to calculate the nominal matrices M₀(q), C₀(q,), F₀(), and G₀(q), are shown in Table I.

The initial position for the experiment with configuration AAA is defined as q(0) = [0, 0, 0]^o and the set-point defined as q(T) = [20, 20, 20]^o, where the vector T = [T₁ T₂ T₃] contains the time duration for the reference trajectory for each joint. This vector is adequately selected taking into account the difference between the initial and final positions. The reference trajectory, q^d, is a fifth-degree polynomial trajectory.

The desired level of attenuation selected for the fully actuated case is g = 3 with the following weighting matrices

Applying the design algorithm described in Section 4, since all conditions are satisfied, one can obtain

The experimental results: joint positions, joint velocities and applied torques, for the configuration AAA, with T = [4.0 4.0 4.0] sec., are shown in Figures 3, 4, and ⁵, respectively. In the following graphics the solid line represents the joint 1, the dashdot line represents the joint 2 and the dashed line represents the joint 3.

The configurations APA and PAP were used to validate the extension of the robotic H_¥ control for underactuated manipulators. For the configuration APA, two control phases are necessary to control all joints to the set-point. Since the configuration APA has n_a = 2 and n_p = 1, we can use two ways to select the controlled joints in the first control phase: 1) the passive joint and one active joint; and 2) only the passive joint, considering the actuation redundant control.

Case 1: In the first control phase, the vector of controlled joints, q_c, is selected as q_c = [q₂ , q₃], i.e., the passive (2) and the active (3) joints are selected (possibility 2 described in Section 3). In the second control phase, the active joints are selected to form the vector of controlled joints, q_c= [q₁ , q₂] (possibility 3). In this phase the passive joint 2 is kept locked, since it has already reached the set-point.

The initial position is q(0) = [0, 0, 0]^o , the set-point is q(T_c, T_a) = [20, 20, 20]^o. Two vectors of time duration, T_c = [T₂T₃] and T_a= [T₁T₃], related with each control phase, have to be constructed to the underactuated case. Exogenous disturbances, starting at t = 0.3 sec, are introduced in the active joints 1 and 3 in the form

respectively. These disturbances are shown in Figure 6, where the solid line represents the disturbance in the joint 1 and the dashed line the disturbance in the joint 3.

The desired level of attenuation is defined as g = 4 and the weighting matrices are given by

and based on Section 4,

For the second control phase the desired level of attenuation is g = 4.5. The weighting matrices and T₀ are given by

and

The experimental results: joint positions, joint velocities and applied torques, for the configuration APA, with T_c = [1.0 1.0] sec. and T_a = [4.0 4.0] sec., are shown in Figures 7, 8, and ⁹, respectively.

Case 2: In the first control phase, the vector of controlled joints, q_c, is selected as q_c = q₂ , i.e., only the passive joint (2) is selected. Considering the actuation redundant control, the remaining joints are the two actives ones, q_r = [q₁ , q₃]. Here, the arbitrary number z is set to zero. The second control phase is the same as in case 1.

The desired level of attenuation, for the first control phase, is defined as g = 4 and the weighting matrices are given by

Applying again the design algorithm described in Section 4, one can obtain

The experimental results: joint positions, joint velocities and applied torques, for the configuration APA with actuation redundant control, and with T_c= [1.0] sec. and T_a = [4.0 4.0] sec., are shown in Figures 10, 11, and ¹², respectively.

For the PAP configuration, three control phases are necessary to control all joints to the set-point. In the first phase, the vector of controlled joints, qc, is selected as qc = q3 (possibility 1 described in Section 3). In the second phase, qc = q1 (possibility 1). And finally, the active joint, qc = q2, is controlled (possibility 3). In the first phase, the joint 1 is kept locked since it is not been controlled. Between the first and the second control phases the joint 2 is repositioned in order to obtain the necessary workspace to control the joint 1. The vector of time duration is defined as T = [T1 Tad T2 T3], where Ti is the time duration of phase i and Tad is the additional time to reposition the joint 1.

For the first control phase g = 5,

and

For the second control phase g = 5,

and

Finally, for the third control phase g = 3,

and

The experimental results: joint positions, joint velocities and applied torques, for the configuration PAP, with T = [1.0 4.0 0.7 3.0] sec., are shown in Figures 13, 14, and ¹⁵, respectively.

6 CONCLUSION

It was presented in this paper the directives to solve the robotic H_¥ control problem for underactuated manipulators. Since the M_cc(.) and C_cc(.,.) matrices from (16) are formed by components of M(.) and C(.,.) matrices, respectively, keeping their proprieties, the solution for the underactuated problem is equivalent to the fully actuated case. The experimental results presented in this paper validate the proposed H_¥ controller for fully actuated and underactuated manipulators. The application of linear parameter varying (LPV) techniques (Huang and Jadbabaie, 1998) to solve the robotic H_¥ control problem is of author's interest for further works. The LPV methodology is used to solve nonlinear matrix inequalities (NLMI) generated by convex characterization of the nonlinear H_¥ control (Lu and Doyle, 1995).

Artigo submetido em 31/10/01

1a. Revisão em 22/01/03; 2a. Revisão em 10/06/03

Aceito sob recomendação do Ed. Assoc. Prof. Liu Hsu

and

where m_i , l_i , lc_i, and I_i are the mass, length, center of mass and inertia of the i-th link and sin_i = sin(q_i), sin_ij = sin(q_i +q_j), cos_i = cos(q_i), cos_ij = cos(q_i +q_j), and cos_ijk = cos(q_i +q_j+q_k).

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APPENDIX A 

), and

The matrices M(q), C(q,), and G(q) for a 3-link planar manipulator with revolute joints, are given by

Publication Dates

• Publication in this collection
  23 Nov 2004
• Date of issue
  Sept 2004

History

• Received
  31 Oct 2001
• Reviewed
  22 Jan 2003
• Accepted
  10 June 2003