Algorithm 1 Optimization process of particle swarm algorithm

Input: Velocity and friction data pair sequence ${\left\{{\left({\overline{\dot{q}}}_i,{\overline{\tau }}_{f,i}\right)}_n\right\}}_{n=1}^N$ Output: Optimal friction parameter set $f_{c,i},f_{s,i},v_{s,i},{\xi }_i,f_{v,i},f_{p,i}$

1.   Initialize iterations t = 0, maximum iterations T 2:   Initialize position permissible range xmin, xmax, Initialize velocity permissible range vmin, vmax 3:   for particle i = 1 to N do 4:       for dimension d = 1 to 6 do 5:         Initialize position xid and velocity vid randomly within permissible range 6:       end for 7:   Initialize particle optimal position with xi 8:   Calculate particle i fitness value 9:   end for 10:   Initialize global optimal position with the particle with the greatest fitness 11:   for t = 0 to T do 12:       for particle i = 1, N do 13:         Calculate particle velocity according to $v_n^{t+1}=\omega v_n^t+c_1{k}_1\left(p_n^t-x_n^t\right)+c_2{k}_2\left(p_g^t-x_n^t\right)$ 14:         Update particle position according to $x_n^{t+1}=x_n^t+\alpha v_n^{t+1}$ 15:         Update inertia factor ${\omega }^t={\omega }_e+\left({\omega }_s-{\omega }_e\right)\left(T-t\right)/T$ 16:         if position beyond permissible range then 17:           Restores position in last iteration, Inverse velocity, Update position according to new velocity 18:         end if 19:         Calculate particle i current fitness value 20:         if the fitness value is better than $p_n^t$ in history then 21:           Set current fitness value as the $p_n^{t+1}$ 22:         end if 23:       end for 24:       if the optimal fitness value is better than optimal particle in history then 25:         Set optima fitness value as the $p_g^{t+1}$ 26:       end if 27:       if the optimal fitness value is less than the threshold twice break 28:   end for 29:   Save optimal fitness value as the optimal friction parameter set result