Tham khảo tài liệu 'control of robot manipulators in joint space - r. kelly, v. santibanez and a. loria part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 72 3 Robot Dynamics q q qTM q q where M q is a matrix of dimension n X n referred to as the inertia matrix. M q is symmetric and positive definite for all q e IR . The potential energy U q does not have a specific form as in the case of the kinetic energy but it is known that it depends on the vector of joint positions q. The Lagrangian q q given by Equation becomes in this case q q -W q . With this Lagrangian the Lagrange s equations of motion may be written as d d dt dq 2 ả I lo dq - l dlqq On the other hand itholhsthat d dtt d Id dt dq 2 ft1ThI q q TM q q en q q M q q M q q. 3T6 o . Considering these expressions the equation of motion takes the form M q n nd qhs -k-rad sTMqU n dU h T 2 ởq ởq or in compact form M q q C q q q ỡ q T where s q Miq q- qlMiq q J q n tr dU0ri 3 17 Q dq Equation 3 IS is the dynamic equal on for robots ot DOF Notice that 3 IS is a nonlmoar vnntonal differential of the state ó I ơ q is a vecton of dimension r called tine vector of centrifugal and Coriolis forces q q is a vector of dimension n of gravitational forces or torquea and Tt ha vector of dimension n called the vector of external Dynamic Model in Compact Form 73 forces which in general corresponds to the torques and forces applied by the actuators at the joints. The matrix C q q e IR X called the centrifugal and Coriolis forces matrix may be not unique but the vector C q q q is indeed unique. One way to obtain C q q is through the coefficients or so-called Christoffel symbols of the first kind Cjjfe q defined as a - 1 dMkj q dMki q dqj dMij q dqk Here Mij q denotes the iyth element of the inertia matrix M q . Indeed the Arjth element of the matrix C q q Ckj q q is given by we do not show here the development of the calculationsto obtainsuchexpressions theinterested reader is invited to see the texts cited at the end of the chapter CkjRq q h Ctf 3 c2jfe z cnjk sù T The model may be viewed as a dynamic system with input .