CQ4

Oct. 15

Q1

Consider the following manipulator that consists of 2 revolving joints and 1 prismatic (sliding) joint. How many degrees of freedom does the structure have?

Answer

3

Q2

• At time t = tk-1, the joint euler angles at Pose 1 are $θ(t_{k-1}) = [θ1, θ2, θ3]^T$
• At time t = tk, the joint euler angles at Pose 2 are θ(tk) = [θ'1, θ'2, θ'3]T

(each euler angle $θ_i = [ θ_{ix}, θ_{iy}, θ_{iz}]^T$)

1. With θ(tk-1) and θ(tk) known, show how you would compute $\dot\theta(t_k)$ using a backwards difference approach where the time step is deltaT.
2. With $\dot\theta(t_k)$ now known, and assuming a Z->Y->X order of rotation, show how you would compute ωroot(tk), ωhip(tk), and ωknee(tk).
3. With all the $ω(t_k)$ now known and given the distance vectors as stated before, show how you would compute the velocity of the ankle joint $v_{ankle}(t_k)$ at time $t = t_k$.

---

Answer

1

$\dot{\theta}(t_k)=\frac{\theta(t_k)-\theta(t_{k-1})}{\Delta T}$

2

$\omega_{\text{root}}(t_k) = L_{root}(\theta_x,\theta_y,\theta_z) \cdot \dot{\theta}_{\text{root}}(t_k)$

$\omega_{\text{hip}}(t_k) = L_{hip}(\theta_x,\theta_y,\theta_z) \cdot \dot{\theta}_{\text{hip}}(t_k)$

$\omega_{\text{knee}}(t_k) = L_{knee}(\theta_x,\theta_y,\theta_z) \cdot \dot{\theta}_{\text{knee}}(t_k)$

3

$v_{ankle}=(\dot F^{root}_{hip}F^{hip}_{knee}F^{knee}_{ankle}+F^{root}_{hip}\dot F^{hip}_{knee}F^{knee}_{ankle}+F^{root}_{hip}F^{hip}_{knee}\dot F^{knee}_{ankle})\begin{bmatrix}0\\0\\0\\1\end{bmatrix}$