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An actuator space optimal kinematic path tracking framework for tendon-driven continuum robots: Theory, algorithm and validation

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## An actuator space optimal kinematic path tracking framework for tendon-driven continuum robots: Theory, algorithm and validation

<div align="right">Ke Qiu, Hongye Zhang, Jingyu Zhang, Rong Xiong, Haojian Lu, Yue Wang State Key Laboratory of Industrial Control and Technology, Zhejiang University</div>

This repository implements the algorithms presented in our article. If you enjoy this repository and use it, please cite our paper.

Our previous work (title: An Efficient Multi-solution Solver for the Inverse Kinematics of 3-Section Constant-Curvature Robots) appears in proceedings of Robotics: Science and Systems 2023.

```
@INPROCEEDINGS{Qiu-RSS-23, 
    AUTHOR    = {Ke Qiu AND Jingyu Zhang AND Danying Sun AND Rong Xiong AND Haojian LU AND Yue Wang}, 
    TITLE     = {An Efficient Multi-solution Solver for the Inverse Kinematics of 3-Section Constant-Curvature Robots}, 
    BOOKTITLE = {Proceedings of Robotics: Science and Systems}, 
    YEAR      = {2023}, 
    ADDRESS   = {Daegu, Republic of Korea}, 
    MONTH     = {July}, 
    DOI       = {10.15607/RSS.2023.XIX.091} 
} 
```

### Demo

**(main_demo.m)** Results of multiple solutions obtained by our algorithm.

**(main_demo2.m)** Results of tracking a straight line path in two different configurations obtained by our algorithm.

### Package overview

**Demo (2 files)**

- main_demo.m 
- main_demo2.m

**Solver (7 files)**  
public:

- micsolver.m  
- micsolverd.m

private: 
- rho.m 
- soln2xi.m 
- get_err.m 
- solve_r1.m 
- solve_r2.m

**Planner (2 files)** 
public: 
- dp.m 
- allocate_time.m

**Numerical methods (5 files)**  
public:

- revise_grad.m  
- revise_dls.m  
- revise_newton.m

private: 
- revise_plot.m 
- jacobian3cc.m

**Quaternion operations (3 files)**

- up_plus.m 
- up_oplus.m 
- up_star.m

**Lie algebra operations (4 files)**

- up_hat.m 
- up_vee.m 
- exphat.m 
- veelog.m

**Conversions (6 files)** 
- arc2q.m 
- q2arc.m 
- arc2xi.m 
- xi2arc.m 
- xi2len.m 
- q2rot.m 
- rot2q.m

**Other tools (6 files)**

- circles3.m 
- circles3c.m 
- frame.m 
- get_end.m 
- collision_indicator.m 
- collision_marker.m

### Functions

#### Solver

**micsolver.m**

`MICSOLVER` Multi-solution solver for the inverse kinematics of 3-section constant-curvature robots.

`[NOS, NOI] = MICSOLVER(L1, L2, L3, Q, R, TOL, IS_A_SOL)` returns the result of solving the 3-section inverse kinematics problem. The function uses preset resolutions or numerical methods to address the inverse kinematics problem. The function exits after one solution is found.

*Input parameters*  
`L1, L2, L3` section length  
`Q, R` desired end rotation and translation  
`TOL` error tolerance  
`IS_A_SOL` function handle  
It is used to judge if the given parameter is a solution to the inverse kinematics problem. The function has two inputs `(ERR, XI)` and one output in boolean type.

*Output parameters*  
`NOS` number of solutions  
`NOI` number of iterations in numerical correction

*Example*
```matlab
L1 = 1; L2 = 1; L3 = 1;
xi = arc2xi(L1, L2, L3, pi.*[1,2,1,2,1,2].*rand(1, 6));
T = get_end(L1, L2, L3, xi);
q = rot2q(T(1:3, 1:3));
r = T(1:3, 4);
tol = 1e-2; fun = @(e, x) e < tol;
tic;
[nos, ~] = micsolver(L1, L2, L3, q, r, tol, fun);
rt = toc*1000;
if nos
 fprintf('A solution is found in %.2f ms.\n', rt);
end
```

**micsolverd.m**

`MICSOLVERD` Multi-solution solver (debug) for the inverse kinematics of 3-section constant-curvature robots.

`[SOL, NOS, NOI] = MICSOLVERD(L1, L2, L3, Q, R, PAR, NOC, TOL, MSTEP, IS_A_SOL, PLOT_EC, PLOT_IT)` returns the result of solving the 3-section inverse kinematics problem.

*Input parameters*  
`L1, L2, L3` section length  
`Q, R` desired end rotation and translation  
`PAR` length of the partition  
Scalar. The interval $[0, 1)$ is partitioned into subintervals with length `PAR`. A smaller `PAR` means finer resolution. We recommend `PAR = 0.03` for better efficiency and `PAR = 0.01` for more solutions.  
`NOC` number of corrections  
A 1-by-2 array. The model parameters $\hat{\boldsymbol{r}}_3$ and hat $\hat{\boldsymbol{r}}_1$ are corrected for `NOC(1)` and `NOC(2)` times after the approximation, respectively. Large `NOC` provides closer initial value and more computations. We recommend `NOC = [1, 2]` for better efficiency and `NOC = [5, 5]` for better estimation.  
`TOL` error tolerance  
`MSTEP` allowed maximum steps of iterations  
`IS_A_SOL` function handle  
This function is used to judge if the given parameter is a solution to the inverse kinematics problem. The function has two inputs `(ERR, XI)` and one output in boolean type.  
`PLOT_EC` set `'plot'` to visualise the traversal  
`PLOT_IT` set `'plot'` to visualise the numerical correction

*Output parameters*  
`SOL` solutions  
A 6-by-`NOS` array. Each column is the overall exponential coordinate `XI`.  
`NOS` number of solutions  
`NOI` number of iterations in numerical correction

*Example*
```matlab
L1 = 1; L2 = 1; L3 = 1;
alpha = 15*pi/16; omega = [0.48; sqrt(3)/10; -0.86];
q = [cos(alpha/2); sin(alpha/2)*omega];
r = [-0.4; 1.1; 0.8];
tol = 1e-2; fun = @(e, x) e < tol;
[sol, ~, ~] = micsolverd(L1, L2, L3, q, r, ...
 0.01, [5, 5], tol, 10, fun, ...
 'plot', 'plot');
for eta = 1: size(sol, 2)
 fh = figure();
 circles3(fh, L1, L2, L3, sol(:, eta), 'k-');
 view(119, 20);
end
```

**rho.m**

`RHO` Computes the linear distance between two ends of a circular arc.