SCIENTIA SINICA Informationis, Volume 50 , Issue 5 : 718-733(2020) https://doi.org/10.1360/SSI-2019-0094

## Position and posture control for a planar underactuated manipulator based on model reduction and chained structure

• AcceptedOct 8, 2019
• PublishedApr 23, 2020
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### Abstract

A control strategy is proposed based on the model reduction and the chained structure for a planar four-link (Active-Active-Passive-Active, AAPA) underactuated manipulator to achieve its position-posture control objective. The whole control process is divided into three stages. In the first stage, the planar AAPA manipulator is reduced to the planar virtual three-link (Active-Active-Passive, AAP) manipulator by controlling the angle of the fourth link and rotating it to zero. In the second stage, the model of a planar virtual AAP manipulator is transformed into the standard chain structure form. Then, the corresponding controllers are designed to control the passive joint of the planar virtual AAP manipulator to its target position, and the posture angle of the passive link is controlled to its middle posture angle at the same time. At the end of this stage, the planar AAPA manipulator is reduced to the planar Acrobot. In the third stage, the angle of the active link of the planar Acrobot is controlled to its target angle. Also, the angle control of the passive link of the system is realized. Consequently, the position-posture control objective of the planar AAPA manipulator is recognized. Considering the angle constraint of the planar Acrobot, the genetic algorithm is used to coordinate and optimize the target angle of the passive joint, the middle posture angle of the passive link, the target angle of the fourth link, and the target posture angle of the passive link. These ensure the target angles of the planar Acrobot corresponding to the target position-posture of the system can be found. Finally, simulation results demonstrate the effectiveness of the proposed control strategy.

### References

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• Figure 1

(Color online) The model of the planar AAPA manipulator

• Figure 2

(Color online) The model of the planar virtual AAP manipulator

• Figure 3

(Color online) The model of the planar Acrobot

• Figure 4

Multiple target angles corresponding to a target position-posture

• Figure 5

(Color online) Simulation results for case 1. (a) Angles of links; (b) angles velocities of links; (c) coordinate of the passive joint; (d) coordinate of the system endpoint; (e) control torques; (f) posture angles of the passive link and the fourth link

• Figure 6

(Color online) Simulation results for case 2. (a) Angles of links; (b) angles velocities of links; (c) coordinate of the passive joint; (d) coordinate of the system endpoint; (e) control torques; (f) posture angles of the passive link and the fourth link

• Table 1   The model parameters of the planar AAPA manipulator
 Segment $i$ ${m_i}\left(~{{\rm{kg}}}~\right)$ ${l_i}\left(~{\rm{m}}~\right)$ ${l_{ci}}\left(~{\rm{m}}~\right)$ ${J_i}\left(~{{\rm{kg/}}{{\rm{m}}^{\rm{2}}}}~\right)$ 1 1.2 1.2 0.144 0.144 2 1.2 1.2 0.144 0.144 3 0.6 0.3 0.144 0.0022 4 0.6 0.7 0.144 0.0285
•

Algorithm 1 Optimization of each target value of the system

Set up the parameters of GA: ${p_s},{p_c},{p_m},N,G,{N_{{\rm{var}}}}~=~4,\Omega~~\in~\left\{~{~-~2\pi~,2\pi~}~\right\}$;

Randomly initialize: ${P_k}(g)~=~\left\{~{{x_{pc}},{y_{pc}},{\theta~_{pc}},{q_{4c}}}~\right\}~\in~\Omega,\left(~{k~=~1,2,3,~\ldots~,N}~\right)$;

for $g=1:G{\rm{~+1~}}$

Substituting ${q_{4c}}$ and ${\theta~_{pc}}$ into (6), the posture angle ${\theta~_{pcd}}$ of the passive link is obtained;

Substitute (2), (6) and ${\theta~_{pcd}}$ into (7), and calculate $h$;

if $h~<~\varepsilon~$ then

${x_{pd}}~=~x_{pc}^k,{y_{pd}}~=~y_{pc}^k,{\theta~_{pd}}~=~\theta~_{pc}^k,{\theta~_{pdd}}~=~\theta~_{pcd}^k,{q_{4d}}~=~q_{4c}^k$;

$\bf{{break}}$;

end if

Update ${x_{pc}},~{y_{pc}},~{\theta~_{pc}},~{\theta~_{pcd}}$ and ${q_{4c}}$ through crossover, mutation, selection operations;

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