The nonlinear dynamic equations

  The nonlinear dynamic equations for an m-link robot are given as 𝑀(𝑞)𝑞̈+𝐶(𝑞,𝑞̇)𝑞̇ +𝐷𝑞̇ +𝑔(𝑞)=𝑢 where 𝑞 is an m-dimensional vector of generalized coordinates representing joint positions, 𝑢 is an m-dimensional control torque input, and 𝑀(𝑞) is a symmetric inertia matrix, which is positive definite for all 𝑞𝜖𝑅𝑚 . The term 𝐶(𝑞,𝑞̇)𝑞̇ accounts for centrifugal and Coriolis forces. The matrix 𝐶 has the property that 𝑀̇ −2𝐶 is a skew-symmetric negative definite matrix for all 𝑞̇𝜖𝑅𝑚, i.e., 𝑞̇𝑇(𝑀̇ −2𝐶)𝑞̇ =0, where 𝑀̇ is the derivative of 𝑀(𝑞) with respect to time 𝑡. The term 𝐷𝑞̇ accounts for vicious damping, where 𝐷 is a positive semidefinite symmetric matrix. The term 𝑔(𝑞) accounts for gravity forces. (a) When 𝑢=𝑔(𝑞), prove that the system is stable using direct Lyapunov method. (b) When 𝑢=𝑔(𝑞)−𝐾𝑝(𝑞−𝑞∗)−𝐾𝑑𝑞̇ , where 𝐾𝑝 and 𝐾𝑑 are positive diagonal matrices and 𝑞∗ is a desired fixed position in 𝑅𝑚, find the equilibrium points of the closed-loop system and prove that it is asymptotically stable. (c) Assume that 𝑀(𝑞), 𝐶(𝑞,𝑞̇), 𝐷, 𝑔(𝑞) are known parameters and 𝑞𝑑 is a desired trajectory in 𝑅𝑚. Design a sliding mode controller to achieve trajectory tracking and verify the system stability. (d) Assume that 𝑀, 𝐶, 𝐷, 𝑔 are unknown constants and 𝑞𝑑 is a desired trajectory in 𝑅𝑚. Design an adaptive controller to achieve trajectory tracking and verify the system stability. (e) Assume that 𝑀(𝑞), 𝐶(𝑞,𝑞̇), 𝐷, 𝑔(𝑞) are unknown functions of 𝑞 and 𝑞̇. 𝑞𝑑 is a desired trajectory in 𝑅𝑚. Design an adaptive neural network controller to achieve trajectory tracking and verify the system stability