Modeling, optimizing and simulating robot calibration with accuracy improvement

This work describes techniques for modeling, optimizing and simulating calibration processes ofrobots using off-line programming. The identification of geometric parameters of the nominalkinematic model is optimized using techniques of numerical optimization of the mathematicalmodel. The simulation of the actual robot and the measurement system is achieved by introducingrandom errors representing their physical behavior and its statistical repeatability. An evaluationof the corrected nominal kinematic model brings about a clear perception of the influence ofdistinct variables involved in the process for a suitable planning, and indicates a considerableaccuracy improvement when the optimized model is compared to the non-optimized one

Significance of Observation Strategy on the Design of Robot Calibration Experiments

In this paper, it is proved that a trajectory tracking system of a manipulator is globally stable if the system is controlled under the decentralized PD control law plus a sliding term with a constant coefficient, and the norm of the coefficient matrix of its differential term is no less than that of the centripetal and Coriolis' force term corresponding to the desired angular velocity, i.e., \\K d \\ > HC(q, q rf )ll. Condition \\K d \\ > IIC(q, q d )ll implies that K d increases only with q rf instead" of q. A type of globally asymptotically stable adaptive sliding mode PD-based control scheme is proposed, and the proof of stability of the system is also given. It is easy to implement in real-time compared with other adaptive control laws as no estimation of gravitational and frictional forces is necessary. Introduction It is one of the challenging problems in robotics to control robot manipulators of high nonlinearity and coupling with high precision and high speed, which has received considerable attention. In recent years, many control laws for manipulators have been proposed. These control laws can be roughly classified into linear and nonlinear model-based control laws. • There are many linear control laws for robot manipulators, such as the decentralized linear PID law widely used in industrial robots, the optimal control law Many nonlinear model-based control laws have been proposed, such as the computed torque method In this paper, it is proved that a trajectory tracking system of a manipulator is globally stable if the system is controlled under the decentralized PD control law plus a sliding term with a constant coefficient, and the norm of the coefficient matrix of its differential term is no less than that of the centripetal and Coriolis' force term corresponding to the desired angular velocity, i.e., \\K d \\ > HC(q, q rf )ll. Condition \\K d \\ > IIC (q, q f/ )ll implies that K d increases only with q d instead of q. A type of globally asymptotically stable adaptive sliding mode PD-based control scheme is proposed, and the proof of stability of the system is also given. It is easy to implement in real-time compared with other adaptive control laws since estimation of gravitational and frictional forces is not required. Mathematical Description For a robot manipulator with a set of n rigid links, if the joint angular variables are identified by q = [q\, T , the dynamics of the manipulator can be expressed as follows where M(q) e R" x " is a positive definite symmetrical inertia matrix of the manipulator, g(q) e 7?" xl a gravitational force vector, f(q, T) £ R" XI a frictional force vector, T ei?" xl a joint torque vector applied by the actuators fixed on the manipulator, and C(q, q)q € R" then (2.1) can be written as 2) By premultiplying both sides of (3.2) by e r , taking the integral of both sides, and by recalling the skew-symmetric nature of matrix M(q) -2C(q, q) [9-11], we get = -e J (t i )M(q(t l ))e(t i )--e'(t 0 )M(q(t 0 ))e(t Q ) >--e From (3.3), we can see that if e T is used as an output vector of a system and the right hand side of (3.2) is taken as an input vector of the system, then the error system defined by (3.2) is a positive mechanical system Therefore, for the trajectory tracking system defined by (2.1) and controlled under the decentralized PD-based control law plus a sliding term, the following results can be obtained: and the coefficient matrix of its differential term satisfies yUI>IIC(q,q rf )ll (3.5) then the system defined by (2.1) is globally stable, where 11-11 denotes the norm of a matrix, and Proof: From (3.2) and (3.4), an equivalent block diagram of the system can be built as shown in ri=-Ape (3.8) ? = Me + C(q,q)e (3.10) It can be seen from the block diagram that the equivalent system consists of a linear block and a nonlinear block. The linear block (also called a feedforward block) is a vector integrator with its input e and integral coefficient K p . The nonlinear block (also referred to as a feedback block) consists of a feedforward part and a feedback part. The input of the feedforward part is -f, and its output is -e. The input of the feedback part is -e, and its output is -A^e -g(q) -f(q, T) -M(q)q f/ -C(q, q)q rf -c x sign(e). The linear block satisfies the positive real condition. From (3.3), the feedforward part of the feedback block satisfies Popov's integral inequality. By use of the properties of a positive system Hence, the system is globally stable. Adaptive Sliding Mode Control Scheme In the preceding section, the error vector e is taken as a state vector of the equivalent system to prove that the system is globally stable. It is necessary that the linear (feedforward) block is strictly positive real to prove that a control system is globally asymptotically stable on the basis of Hyperstable Theory e r (t) = q(t)-q r (t), e r {t) = e (0 + Ae(t) (4.2) where A = diag[X,, X 2 , ..., X"], A, > 0, / = 1, 2, ..., n. With these symbols, the dynamics of a manipulator can be rewritten as follows Proof: The proof of Corollary 4.1 is much the same as Theorem 3.1. By use of (4.1)-(4.2), we can rewrite (2.1) as follows M(q)e r +C(q,q)e f =T-Mq r -C(q,q)q r -g(q)-fc(q,T)~//q (4.8) From (4.4)-(4.6) and (4.8), an equivalent block diagram of the system can be constructed in the same way as The integral of the second term of (4.9) is, p'i c'i p'i e A 7/e*= e T f,edt+\ e T kf,tdt Finally, if (4.7) is satisfied, the integral of the third term of (4.9) is, ej[c 2 sign(e r ) + g(q) + f c (q,T)+/,q rf ]> -7 5 (4.12) Then from (4.10)-(4.12), we can conclude that the nonlinear (feedback) block of the equivalent system satisfies Popov's integral inequality. This, in conjunction with the strictly positive real linear (feedforward) block of the equivalent system, ensures that the system is globally asymptotically stable, and 0 is bounded. Conclusion In this paper, we have proved that a trajectory tracking system of a manipulator is globally stable if the system is controlled under the decentralized PD control law plus a sliding term with a constant coefficient, and the norm of the coefficient matrix of its differential term is no less than that of the centripetal and Coriolis' force term corresponding to the desired angular velocity, that is, IIA^II > IIC(q, qd)ll. Condition \\K d \\ > IIC(q, q rf )ll implies that K d should be increased only with i\ d instead of q in order to guarantee stability of the system, which is active to control a robot manipulator by the feedback control schemes. At the same time, condition IIA^II > llC(q, q d )ll reveals an essential distinction between the nonlinear system and a linear system where K d is a constant matrix. The sliding term rejects bounded uncertainties of the system. In Corollary 4.1, although the system is globally asymptotically stable under the control of the adaptive sliding mode PD-based control scheme, it is easy to excite unmodeled highfrequency modes of the system and cause chattering. If e r / IIM + e (e > 0) is substituted for sign(e r ), chattering of the system can be reduced. However, there is a steady-state error proportional to t. Substituting the proportional adaptive control scheme for the proportional and integral adaptive control scheme, though simple, lowers the adaptive rate of the system. References 1 Takegati, M., and Arimoto, S., "A New Feedback Method for Dynamic Control of Manipulators," ASME JOURNAL OF DYNAMIC SYSTEMS, MEASURE-MENT, AND CONTROL, Vol. 102, June, 1981, pp. 119-125. 2 Arimoto, S., and Miyazaki, F., "Stability and Robustness of PID Feedback Control for Robot Manipulator of Sensory Capability," Proc. 1st Int. Symp. Robotics Res., MIT Press, Cambridge, MA, 1983, pp. 784-799. 3 Kawamura, S., Miyazaki, F., and Arimoto, S., " AC-30, No. 12, Dec. 1985, pp. 1229-1233 7 Bejczy, A. K., "Robot Arm Dynamics and Control," JPL NASA Technical Memorandum 33-669, Feb. 1974. 8 Young, K. K. D., "Controller Design for a Manipulator Using Theory of Variable Structure Systems," IEEE Trans. System, Man, Cybernetics, Vol. SMC-8, No. 2, Feb. 1978, pp. 101-109. 9 Craig, J. J., Hsu, P., and Sastry, S. S., "Adaptive Control of Mechanical Manipulators, " Proc. 1986 IEEE Int. Conf. Robotics Automat., Apr. 1986 10 Slotine, J. J., and Li, W., "On the Adaptive Control of Robot Manipulators," Int. J. Robotics Res., Vol. 6, No. 3, Fall 1987, pp. 49-59. 11 Sadegh, N., and Horowitz, R., "Stability Analysis of an Adaptive Controller for Robotic Manipulators," Proc. 1987 / Vol. 115, SEPTEMBER 1993 Transactions of the ASM