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Control of Redundant Robot Manipulators - R.V. Patel and F. Shadpey Part 7

Tham khảo tài liệu 'control of redundant robot manipulators - r.v. patel and f. shadpey part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.2 Literature Review 81 4.2 Literature Review 4.2.1 Constrained Motion Approach This approach considers the control of a manipulator constrained by a rigid object1 in its environment. If the environment imposes purely kinematic constraints on the end-effector motion only a static balance of forces and torques occurs assuming that the frictional effects are neglected . This implies no energy transfer or dissipation between the manipulator and the environment. This underlies the main modeling assumption made by 45 where an algebraic vector equation restricts the feasible end-effector poses. The constrained dynamics can be written as Hq q h q q T JTF p 0 4.2.1 where T is the vector of applied forces torques H q is the n X n symmetric positive definite inertia matrix h is the vector of centrifugal Coriolis and gravitational torques. p e Rn is the generalized task coordinates and O p e Rm is the constraint equation continuously differentiable with respect to p. It is assumed that the Jacobian matrix is square and of full rank. The analysis given below follows that in 45 the generalized force2 F in 4.2.1 is given by U-IU-- Tx õp F 4.2.2 where X e Rm x 1 is the vector of generalized Lagrange multipliers. Using the forward kinematic relations p Jq p Jq Jq 4.2.3 1. A work environment or object is said to be rigid when it does not deform as a result of application of generalized forces by the manipulator. 2. In the rest of this chapter the term force refers to both interaction force and torque. 82 4 Contact Force and Compliant Motion Control and assuming that the Jacobian matrix is invertible we can obtain the following constrained dynamics expressed with respect to generalized task coordinates directly from 4.2.1 Hip p hp p p u - F 42.4 O p 0 where Hp J-tH q J-1 hp -HpJq J Th q q 4.2.5 u J tx A nonlinear transformation can then be used to transfer to a new coordinate frame. It is assumed that there is an open set 0C Rn - m and a function Q such that where O Q p2 p2 0 Vp2 e