Algorithm 1 HPPM-PRA general procedure.
Require:  $R\left(w\right),C\left(w\right)$, $E{s}_{start},E{s}_{goal}$, $L,v$	▹ The task to be solved is proposed
Require:  $P_C,P_R$	▹ Assign the repulsion parameter
1: Get $(w_{start},w_{goal},a_0)$	▹ See Algorithm 2
2: Get the value of Q	▹ See Algorithm 3
3: Set the non-linear equation system to solve ($f_1,\dots ,f_v$)	▹ See Algorithm 4
4: Generate the homotopy equation (H)	▹ See Algorithm 5
5: Create hypersphere $S_i$	▹$w_{start}$ is used as the center of the first hypersphere
6: Formulation of the homotopy system ($H_S$)
7: iteration=0	▹ A temporary variable is used as the counter
8: while$(\lambda \ne 1)$do	▹ Use the hyperspherical path-tracking algorithm, until $\lambda =1$
9:     if $(iteration<2)$ then
10:         Euler’s predictor ($H_S$)	▹ Euler’s predictor is used
11:     else
12:         Vector predictor ($H_S$)	▹ The vector predictor is used
13:     end if
14:     Broyden’s method ($H_S$)	▹ The corrector method is used
15:     The numeric homotopy path is stored
16:     Update the center of the hypersphere ($S_i$)
17:     iteration++
18: end while
Ensure:  The numerical homotopy path	▹ The robotic arm can execute the path