Tracking Control for Robot Manipulators with Kinematic and Dynamic Uncertainty

Transcription

1 Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seille, Spain, December 2-5, 2005 WeB3.3 Tracking Control for Robot Manipulators with Kinematic and Dynamic Uncertainty D. Braganza, W. E. Dixon, D. M. Dawson, B. Xian Abstract The control objectie in many robot manipulator applications is to command the end-effector motion to achiee a desired response. To achiee this objectie a mapping is reuired to relate the joint/link control inputs to the desired Cartesian position and orientation. If there are uncertainties or singularities in the mapping, then degraded performance or unpredictable responses by the manipulator are possible. To address these issues, an adaptie tracking controller is deeloped in this paper for robot manipulators with uncertainty in the kinematic and dynamic models. The controller is deeloped based on the unit uaternion representation so that singularities associated with three parameter representations are aoided. I. INTRODUCTION The control objectie in many robot manipulator applications is to command the end-effector motion to achiee a desired response. The control inputs are applied to the manipulator joints, and the desired position and orientation is typically encoded in terms of a Cartesian coordinate frame attached to the robot end-effector with respect to the base frame (i.e., the so-called task-spacariables). Hence, a mapping (i.e., the solution of the inerse kinematics) is reuired to conert the desired task-space trajectory into a form that can be utilized by the joint space controller. If there are uncertainties or singularities in the mapping, then this can result in degraded performance or unpredictable responses by the manipulator. Seeral parameterizations exist to describe orientation angles in the task-space to joint-space mapping, including three-parameter representations (e.g., Euler angles, Rodrigues parameters) and the four-parameter representation gien by the unit uaternion. Three-parameter representations always exhibit singular orientations (i.e., the orientation Jacobian matrix in the kinematic euation is singular for some orientations), while the unit uaternion represents the end-effector orientation without singularities. By utilizing the singularity free unit uaternion, the emphasis in this paper is to deelop a tracking controller that compensates for uncertainty throughout the kinematic and dynamic models. Some preious task-space control formulations based on the unit uaternion can be found in, 2, 5, 9 and This work is supported in part by a DOC Grant, an ARO Automotie Center Grant, a DOE Contract, a Honda Corporation Grant and a DARPA Contract at Clemson Uniersity and in part by AFOSR contract number F , a DARPA grant, and by research grant No. US from BARD, the United States - Israel Binational Agricultural Research and Deelopment Fund at the Uniersity of Florida. D. Braganza and D. M. Dawson are with the Electrical and Computer Engineering Dept., Clemson Uniersity, Clemson, SC 29634, USA. W. E. Dixon is with the Mechanical and Aerospace Engineering Dept., Uniersity of Florida, Gainesille, FL 326, USA. wdixon@ufl.edu B. Xian is with the Dept. of Mechanical Engineering and Material Sciences, Duke Uniersity, Durham, NC 27705, USA. 2. A uaternion-based resoled acceleration controller was presented in 2, and uaternion-based resoled rate and resoled acceleration task-space controllers were proposed in 2. Output feedback task-space controllers using uaternion feedback were presented in 5 for the regulation problem and in for the tracking problem. Model-based and adaptie asymptotic full-state feedback controllers and an output feedback controller based on a model-based obserer were deeloped in 9 based on the uaternion parameterization. A common assumption in most of the preious robot controllers (including all of the aforementioned uaternionbased task-space control formulations) is that the robot kinematics and manipulator Jacobian are assumed to be perfectly known. From a reiew of literature, few controllers hae been deeloped that target uncertainty in the manipulator forward kinematics and Jacobian. For example, Cheah et al. 3-8 deeloped seeral approximate Jacobian feedback controllers that exploit a static, best-guess estimate of the manipulator Jacobian to achiee task-space regulation objecties despite parametric uncertainty in the manipulator Jacobian. In 20, Yazarel and Cheah deelop a task-space adaptie controller for set point control of robots with uncertainties in the graity regressor matrix and kinematics. In 0, Dixon deeloped an adaptie regulation controller for robot manipulators with uncertainty in the kinematic and dynamic models. The result in 0 also accounted for actuator saturation since the maximum commanded torue could be a priori determined due to the use of saturated feedback terms in the controller. All of the aforementioned controllers that account for kinematic uncertainty are based on the three-parameter Euler angle representation. Moreoer, all of the preious results only target the set point regulation problem. The only result that targets the more general tracking control problem for manipulators with uncertain kinematics is gien in 9. The result in 9 is also based on the Euler angle representation and the controller reuires the measurement of the task spacelocity. Hence motiated by preious work, an adaptie tracking controller is deeloped in this paper for robot manipulators with uncertainty in the kinematic and dynamic models. The controller is deeloped based on the unit uaternion representation so that singularities associated with three parameter representations are aoided. In addition, the deeloped controller does not reuire the measurement of the task spacelocity. The stability of the controller is proen through a Lyapuno-based stability analysis /05/$ IEEE 5293

2 II. ROBOT DYNAMIC AND KINEMATIC MODELS A six-link, rigid, reolute robot manipulator can be described by the following dynamic model 4 M(θ) θ + V m (θ, θ) θ + G(θ)+F d θ τ. () In (), θ (t) R 6 is the joint position, M (θ) R 6 6 represents the inertia matrix, V m (θ, θ) R 6 6 is the centripetal-coriolis matrix, G (θ) R 6 is the graity ector, F d R 6 6 is a constant diagonal matrix which represents thiscous friction coefficients, and τ (t) R 6 represents the input toruector. The dynamic model gien in () has the following properties 4, which are utilized in the subseuent control design and analysis: Property : The inertia matrix is symmetric and positiedefinite, and satisfies the following ineualities m x 2 x T M(θ)x m 2 x 2 x R 6 (2) where m,m 2 R are positie constants and denotes the standard Euclidean norm. Property 2: The inertia and centripetal-coriolis matrices satisfy the following skew-symmetric relationship ( ) x T 2 Ṁ(θ) V m(θ, θ) x 0 x R 6. (3) Property 3: The centripetal-coriolis matrix satisfies the following skew-symmetric relationship V m (θ, x)y V m (θ, y)x x, y R 6. (4) Property 4: The norm of the centripetal-coriolis matrix and the norm of the friction matrix, can be upper bounded as follows: V m (θ, x) i ζ c x x R 6, F d ζ f (5) where ζ c,ζ f R are positie constants, and i denotes the induced-infinity norm of a matrix. Property 5: Parametric uncertainty in M (θ),v m (θ, θ), G (θ) and F d, is linearly parameterizable. Let E and B be orthogonal coordinate frames attached to the manipulator s end-effector and fixed base, respectiely. The position and orientation of E relatie to B can be represented through the following forward kinematic model 5 p hp (θ) h (θ). (6) In (6), h p ( ) :R 6 R 3 denotes a function that maps θ (t) to the measurable 2 task-space position coordinates of the end-effector, denoted by p ( ) R 3,andh ( ) :R 6 R 4 denotes a function that maps θ (t) to the measurable 3 unit It is assumed that the actuated manipulator joint is rigidly connected to the links, so that the link-space and joint-space are euialent. Hence, the words joint and link can be used interchangeably. 2 The task-space position of E relatie to B is assumed to be measurable, as in 3-8, 9, 0, and 20. For example, a camera system could be utilized. 3 The task-space orientation of E relatie to B is also assumed to be measurable through the use of a camera system. uaternion and is denoted by (t) R 4. The unit uaternion ector, denoted by (t) o (t), T (t) T with o (t) R and (t) R 3 2, 4, proides a global nonsingular parameterization of the end-effector orientation, and is subject to the constraint, T. Seeral algorithms exist to determine the orientation of E relatie to B from a rotation matrix that is a function of θ (t). Conersely, a rotation matrix, denoted by R() SO(3), can be determined from agien(t) by the formula 5 R () ( 2 o T ) I3 +2 T +2 o (7) where I 3 is the 3 3 identity matrix, and the notation a, a a,a 2,a 3 T, denotes the following skew-symmetric matrix a 0 a 3 a 2 a 3 0 a. (8) a 2 a 0 The time deriatie of (6) is gien by the following expression Jp θ (9) J where J p (θ) :R 6 R 3 6 and J (θ) :R 6 R 4 6 denote the unknown position and orientation Jacobian matrices, respectiely, defined as J p (θ) h p / θ and J (θ) h / θ. To facilitate the subseuent deelopment, (9) is expressed as follows: J(θ) ω θ J where J (θ) p 2B T R J 6 6. (0) Remark : The dynamic and kinematic terms for a general reolute robot manipulator, denoted by M(θ), V m (θ, θ), G(θ) and J(θ), areassumedtodependonθ(t) only as arguments of trigonometric functions and hence, remain bounded for all possible θ(t). During the control deelopment, the assumption will be made that if p(t) L, then θ(t) L (Note that (t) is always bounded, since T ). The expression in (0) is obtained by exploiting the fact that (t) is related to the angular elocity of the end-effector, denoted by ω (t) R 3, ia the following differential euation ω B T () where the known Jacobian-like matrix B () :R 4 R 4 3 is defined as follows: B T 2 o I 3. (2) Property 6: The kinematic system in (0) can be linearly parameterized as follows: J θ W j φ j (3) where W j (θ, θ) R 6 n denotes a regression matrix of measurable signals, and φ j R n denotes a ector of n unknown constants. 5294

3 Property 7: There exists upper and lower bounds for the parameter φ j such that J(θ, φ j ) is always inertible. We will assume that the bounds for each parameter can be calculated as follows (4) where, R denotes the i th component of φ j R n and, R denote the i th components of φ j, φ j R n, which are defined as follows T φ j φ j,φ j2,,φ jn φ j (5) T φ j, φ j2,, φ jn. III. PROBLEM STATEMENT The objectie is to the design the control input τ(t) to ensure end-effector position and orientation tracking for the robot model gien by () and (0) despite parametric uncertainty in the kinematic and dynamic models. We will assume that the only measurable signals are the joint position, joint elocity, and end-effector position. To mathematically uantify this objectie, a desired position and orientation of the robot end-effector is defined by a desired orthogonal coordinate frame E d. Thector p d (t) R 3 denotes the position of the origin of E d relatie to the origin of B, while the rotation matrix from E d to B is denoted by R d (t) SO(3). The end-effector position tracking error e p (t) R 3 is defined as e p p d p (6) where p d (t), d (t), and p d (t) are assumed to be known bounded functions of time. If the orientation of E d relatie to B is specified in terms of a desired unit uaternion d (t) od (t), d T (t) T R 4, with od (t) R and d (t) R 3.Thensimilarlyto(7),R d ( d ) can be calculated from d (t) as follows R d ( d ) ( 2 od T d d) I3 +2 d T d +2 od d (7) where it is assumed that R d, Ṙ d, R d L. As in (), the time deriatie of d (t) is related to the desired angular elocity of the end-effector (i.e., the angular elocity of E d relatie to B), denoted by ω d (t) R 3, through the known kinematic euation d B( d )ω d. (8) To uantify the difference between the actual and desired end-effector orientations, we define the rotation matrix R SO(3) from E to E d as follows R Rd T R ( e 2 o e T ) I3 +2 e T +2e o e (9) where e (t) e o (t), e T (t) T R 4 represents unit uaternion tracking error that satisfies the constraint e T e e 2 o + et. (20) The uaternion tracking error e (t) can be explicitly calculated from (t) and d (t) ia uaternion algebra by noticing that the uaternion euialent of R Rd T R is the following uaternion product 3, 2 e d (2) where d (t) od (t), d T (t) T R 4 is the unit uaternion representing the rotation matrix Rd T ( d). Afterusing uaternion algebra, the uaternion tracking error can be deried as follows (see 2 and Theorem 5.3 of 3) eo o od + T d od o d +. (22) d Based on (), (8), and (22), the unit uaternion error system can be formulated as follows ėo ė 2 et 2 (e oi 3 e ). (23) The angular elocity of E with respect to E d with coordinates in E d, denoted by (t) R 3, can be calculated from (9) as follows 8 R T d (ω ω d ). (24) The end effector tracking errors are then written using (0), (6), and (24) as ( ) ėp d Λ + J ω θ (25) d where Λ R 6 6 is defined as I3 0 Λ Rd T, (26) where represents a 3 3 matrix of zeros. Based on the aboe definitions, the tracking objectie defined in terms of the end-effector position and unit uaternion error is to design the control input τ(t) such that e p (t) 0 and (t) 0 as t. (27) The orientation tracking objectie gien in (27) can also be stated in terms of e (t). Specifically, (20) implies that 0 (t) and 0 e o (t) (28) for all time and if (t) 0 as t then e o (t) as t. Thus, if (t) 0 as t then (9) along with the preious statement can be used to conclude that R(t) I3 as t, and hence, the orientation tracking objectie has been achieed. IV. TRACKING ERROR SYSTEM DEVELOPMENT To facilitate the deelopment of the open-loop error system, an auxiliary ariable η(t) R 6 is defined as follows: ( ) η ΛĴ Rd T ω + d + k 2 e θ (29) where k,k 2 R 3 3 are positie, constant, diagonal matrices, and Ĵ ( ) R6 6 is an estimated manipulator Jacobian matrix. After adding and subtracting the terms ΛĴ ( ) θ(t) 5295

4 and ΛĴ ( ) η(t) to (25) and utilizing (26), the following kinematic error system can be deeloped ėp k e p +Λ k 2 (Ĵη + Wj φj ) (30) where W j ( ) R 6 n was introduced in (3) and the parameter estimation error term φ j (t) R n is defined as φ j φ j ˆφ j. (3) The adaptie estimate ˆφ j (t) R n introduced in (3) is designed as follows: ˆφ j proj {y} (32) where the auxiliary term y R n is defined as y Γ Wj T Λ T ep (33) where Γ R n n is a constant positie diagonal matrix and the function proj{y} is defined as follows y i if ˆ > y i if ˆ and y i > 0 0 if proj{y i } ˆ and y i < 0 0 if ˆ and y i > 0 y i if ˆ and y i 0 y i if ˆ < (34) ˆ (0) (35) where y i denotes the i th component of y, and ˆ (t) denotes the i th component of ˆφ j (t) (Note that the aboe projection algorithm ensures that φ j ˆφ j (t) φ j. For further details the reader is referred 6). To obtain the closed loop error system for η(t), wefirst take the time deriatie of (29) to obtain the following expression η d { ( ) } ΛĴ dt Rd T ω + d + k 2 e θ. (36) After pre-multiplying (36) by M (θ), substituting () into the resulting expression for M (θ) θ(t), and utilizing (29), the following simplified expression can be obtained M η V m η + τ + W y φ y (37) where W y (p d, d, p d, d,ω d, ω d,p,,θ, θ) R 6 n2 is a regression matrix of known and measurable uantities, and φ y R n2 is a ector of n 2 unknown constant parameters. The product W y ( ) φ y introduced in (37) is defined as W y φ y M d { ( ) } ΛĴ dt Rd T ω d + k 2 ) +V m (ΛĴ Rd T ω d + k 2 G(θ) F d θ. (38) Based on (37) and the subseuent stability analysis, the control input τ(t) is designed as ( ) T ep τ W y ˆφy k r η ΛĴ. (39) where k r R 6 6 is a constant positie diagonal matrix and ˆφ y (t) R n2 denotes an adaptie estimate which is generated by the following differential expression ˆφ y Γ 2 Wy T η (40) where Γ 2 R n2 n2 is a positie constant diagonal matrix. After substituting (39) into (37), the following closed-loop error system is obtained ( ) T ep M η V m η + W y φy k r η ΛĴ (4) where the adaptie estimation error is defined as φ y φ y ˆφ y. (42) V. STABILITY ANALYSIS Theorem : Gien the robotic system described by (), the control input (39) along with the adaptie laws defined in (32) and (40) guarantee asymptotic regulation of the end-effector position error and the unit uaternion error in the sense that e p (t) 0 as t and (t) 0 as t. Proof: Let V (t) R denote the following non-negatie scalar function V 2 et p e p +( e 0 ) 2 + e T + 2 φ T j Γ φ j + 2 φ T y Γ 2 φ y + 2 ηt Mη. (43) After taking the time deriatie of (43) and utilizing (23), (3) and (42), the following expression is obtained V e T p ė p +( e 0 ) ( e T ) + e T ( e0 I 3 e ) + 2 ηt Ṁη + η T M η φ T y Γ 2 ˆφ y φ T j Γ ˆφ j. (44) Upon further simplification of euation (44) by canceling common terms, and substituting for M η from (4), the following expression for V (t) can be obtained V e T p e T ė p η T k r η η T V m η + η T W y φy η T ( ΛĴ ) T ep + 2 ηt Mη φ T j Γ ˆφ j φ T y Γ 2 ˆφ y. (45) After using Property 3, substituting from (30), (32), and (40) and canceling terms, V (t) can be expressed as V ( ) e T p e T k e p +ΛW k 2 e j φj η T k r η φ j Γ proj{y}. (46) 5296

5 Substituting for y from (33) and using the definition of the projection function, (34), the expression for V (t) can be upper bounded as follows V λ min {k } e p 2 λ min {k 2 } 2 λ min {k r } η 2. (47) where λ min is the minimum Eigenalue of the matrix. The expressions in (43) and (47) can be used to proe that e p (t), (t), η(t), φ j (t), φ y (t) L and that e p (t), (t), η(t) L 2. Using (6) and the assumption that p d (t) L, it is clear that p(t) L. From (3) and (42) it can be concluded that ˆφ j (t), ˆφ y (t) L. Utilizing Property 7, the definition of η(t) in (29) and the fact that e p (t), (t), η(t) L, we can show that θ(t) L. Moreoer, (9), (6) and the fact that J(θ) L can be used to show that (t), ė p (t) L. From (20), (23), (25) and (28) we can show that e 0 (t), ė 0 (t), ė (t) L. From the definition of W j ( ) and W y ( ) in (3) and (38) respectiely and the preceding arguments, it is clear that W y ( ),W j ( ) L. Utilizing (32), (33), (34) and (40), we can show that ˆφ j (t), ˆφ y (t) L. The definition of τ(t) in (39) can be used to show that τ (t) L ; hence, θ(t), θ(t), θ (t) L and from (36) we can conclude that η(t) L. Since ė p (t), ė (t), η(t) L and e p (t), (t), η(t) L 2, Barbalat s Lemma 7 can be used to show that, e p (t) 0, (t) 0 and η(t) 0 as t. VI. CONCLUSION A task-space, adaptie tracking controller for robot manipulators with uncertainty in both the kinematic and the dynamic models was proposed. The controller yields asymptotic regulation of the end-effector position and orientation tracking errors. The adantages of the proposed controller are that singularities associated with the three parameter representation are aoided. 8 C. C. Cheah, S. Kawamura, S. Arimoto, and K. Lee, H Tuning for Task-Space Feedback Control of Robot with Uncertain Jacobian Matrix, IEEE Trans. Automatic Control, ol. 46, pp , Aug C.C.Cheah,C.Liu,J.J.Slotine, ApproximateJacobianAdaptie Control for Robot Manipulators, in Proc. IEEE Int. Conf. Robotics and Automation, New Orleans, La, April 2004, pp W. E. Dixon, Adaptie Regulation of Amplitude Limited Robot Manipulators with Uncertain Kinematics and Dynamics, in Proc. IEEE American Control Conf., Boston, Massachusetts, June 2004, pp W. E. Dixon, A. Behal, D. M. Dawson and S. Nagarkatti, Nonlinear Control of Engineering Systems: A Lyapuno Based Approach, Birkhauser: Boston, P. C. Hughes, Spacecraft Attitude Dynamics, New York, NY: Wiley, J. B. Kuipers, Quaternions and Rotation Seuences, Princeton, NJ: Princeton Uniersity Press, F. Lewis, D. M. Dawson and C. Abdallah, Robot Manipulator Control: Theory and Practice, Marcel Dekker Publishing Co., New York, NY, F. Lizarralde and J. T. Wen, Attitude Control Without Angular Velocity Measurement: A Passiity Approach, IEEE Trans. Automatic Control, ol. 4, pp , Mar, M. Krstic, I. Kanellakopoulus and P. Kokotoic, Nonlinear and Adaptie Control Design, New York, NY, John Wiley & Sons, J. Slotine, and W. Li, Applied Nonlinear Control, Englewood Cliff, NJ: Prentice Hall Inc., M. Spong, and M. Vidyasagar, Robot Dynamics and Control, New York: John Wiley & Sons Inc., B. Xian, M. S. de Queiroz, D. Dawson, and I. Walker, Task-Space Tracking Control of Robot Manipulators ia Quaternion Feedback, IEEE Trans. Robotics and Automation, ol. 20, pp , Feb H. Yazarel, and C. C. Cheah, Task-Space Adaptie Control of Robotic Manipulators with Uncertainties in Graity Regressor Matrix and Kinematics, IEEE Trans. Automatic Control, ol. 47, pp , Sep J. S. C. Yuan, Closed-Loop Manipulator Control Using Quaternion Feedback, IEEE Trans. Robotics and Automation, ol. 4, pp , Aug REFERENCES F. Caccaale, C. Natale, and L. Villani, Task-Space Control Without Velocity Measurements, in Proc. IEEE Int. Conf. Robotics and Automation, Detroit, MI, May 999, pp R. Campa, R. Kelly, and E. Garcia, On Stability of the Resoled Acceleration Control, in Proc. IEEE Int. Conf. Robotics and Automation, Seoul, Korea, May 200, pp C. C. Cheah, M. Hirano, S. Kawamura, and S. Arimoto, Approximate Jacobian Control for Robots with Uncertain Kinematics and Dynamics, IEEE Trans. Robotics and Automation, ol. 9, pp , Aug C. C. Cheah, M. Hirano, S. Kawamura, and S. Arimoto, Approximate Jacobian Control with Task-Space Damping for Robot Manipulators, IEEE Trans. Automatic Control, ol. 49, pp , May C. C. Cheah, S. Kawamura, and S. Arimoto, Feedback Control for Robotic Manipulators with Uncertain Kinematics and Dynamics, in Proc. IEEE Int. Conf. Robotics and Automation, Leuen, Belgium, May 998, pp C. C. Cheah, S. Kawamura, and S. Arimoto, Feedback Control for Robotic Manipulators with an Uncertain Jacobian Matrix, Int. Journal of Robotic Systems, ol. 2, no. 2, pp. 9-34, C. C. Cheah, S. Kawamura, S. Arimoto, and K. Lee, PID Control for Robotic Manipulator with Uncertain Jacobian Matrix, in Proc. IEEE Int. Conf. Robotics and Automation, Detroit, Michigan, May 999, pp