# NOTES on: [Integral Sliding Mode Based Switched Structure Control Scheme for Robot Manipulators](https://www.researchgate.net/publication/327807849_Integral_Sliding_Mode_Based_Switched_Structure_Control_Scheme_for_Robot_Manipulators)
## Author:
Mohammad Saad.
## Abstract
key elements of the proposed scheme are the inverse dynamics based centralized controller and a set of decentralized controllers.**The idea of using ISM, apart from its feature of providing robustness in front of a wide class of uncertainties, is motivated by its capability of acting as a “perturbation estimator”**

**To reduce the chattering effect of SMC, ISMC is used.**

| Decentralised Control | centralised Control |
| -------- | -------- |
| if the aim is to control each joint independently.|if the robot is controlled as a Multi-Input-Multi-Output (MIMO) system.
| used when high performance of velocity and acc. not required.|higher performances in terms of velocity and acceleration are required
| seen as linear and decoupled system, one for each joint.| used when manipulators do not present gear boxes at the joints
| non-linearities and coupling effects amongst joints seen as disturbance on single joint|cant neglect non linearities and coupling between joints.
| less computation and easy to use. | |

## The robot model

$B(q) \ddot q + n(q,\dot q) = \tau$
$n(q,\dot q)=C(q,\dot q)\dot q+F_v \dot q+g(q)$

$B(q) \to$ Inertia Matrix
$C(q,\dot q) \to$ Centrifugal and coriolis torques
$F_v \to$ Viscous Friction Matrix
$g(q) \to$ gravitational torque vector
$\tau \to$ motor torque.

$e_1(t)=q_{ref} -q$
$e_2(t)=\dot q_{ref}-\dot q$

$e_j=[e_{1_{j}} \space e_{2_{j}}]^T$
$e_{1_{j}},e_{2_{j}} \to$ position and velocity error in joint j

## Switched Structured Control


## Decentralised control scheme (MODE-1)
$k_r \space \epsilon R^{n*n} \to$ gear ration matrix.
$k_r q=q_m ; \space \space \space k_r^{-1}\tau=\tau_m; \space \space \space$
$k_r^{-1}F_v k_r^{-1}=F_m \to$ Matrix of viscous friction riffered to motor shaft.

Inertial matrix B(q) is splitted in two parts as nominal component($\bar B$) and unknown term($\triangle B(q)$).
$B(q)=\bar B+\triangle B(q)$

**The Dynamic Model is given by:**
## Centralised Control Scheme (MODE-2)

**Assumption:**
Estimation of B(q) is perfect and quite good estimation of n($q,\dot q$) is available .
$\hat n (q,\dot q) != n(q,\dot q)$
**The Dynamic Model is given by:**
$B(q)\ddot q +n(q,\dot q)=\tau$
Selecting control torque ($\tau$) as:
$\tau = -B(q)v_{cen}+\hat n(q,\dot q)$
$\ddot q=-v_{cen}+B^{-1}(\hat n(q,\dot q)-n(q,\dot q))$
$h_{cen}=\ddot q_{ref}-B^{-1}(\hat n(q,\dot q)-n(q,\dot q))$

## Switching Block
Performance index $P_{sw}(\hat h)$. It depends on uncertainity and coupling effect $\hat h$
if Mode 1 is active and the coupling terms are greater than $\tau _{max}$ (i.e., high velocity and acceleration performance are required), switch to Mode 2. Vice versa, if Mode 2 is active and the uncertain terms are less than $A_{min}$ (i.e., high velocity and acceleration performanceare not required) switch to Mode 1.

## Integral Sliding Mode Control
$v_j (t)=u_j (t)+u_{ISM_j} (t)$
$v_j(t) \to$ Control variable
$u_j(t) \to$ Nominal control(both in the decentralized and the centralized loop is chosen as a Proportional-Derivative (PD) controller)
$u_{ISM_j} (t) \to$ SMC action to reject uncertainities.

### Error Model:
$\dot e_j(t)=A_j e_j(t)+B_j(v_j(t)+h_j(t))$
$A_j$ and $B_j$ can be found out using the error equations mentioned in Centralised and decentralised Control Scheme.
$$A_j=\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}
;
B_j=\begin{pmatrix}
0\\
1
\end{pmatrix}$$

The Integral Sliding manifold is given by:
$\sum_j(t)=\sigma_j(t)+\phi_j(t)$
$\sum_j \to$auxilary Sliding variable.
$\sigma_j=me_{1_j}+e_{2_j} \to$ actual sliding variable(Sliding surface equaton)
$\phi_j \to$ Integral term
$\phi_j(t)=-\sigma_j(t_0)-\int_{t_0}^tme_{2_j}(\zeta)+u_j(\zeta)d\zeta$
Initial Conditiona $\to \phi_j(t_0)=-\sigma_j(e_j(t_0))$
Discontinuous Control law is given by: $u_{ISM_j}(t)=-k_jsgn(\sum_j(t))$

***NOTE:*** **ISM controller is able to estimate the uncertain and coupling terms IF THE EQUIVALENT CONTROL IS AVAILABLE. An approximate of equivalent control can be obtained via a first order linear filter with the real discontinuous control as input signal.**

$\tilde u_{ISM_{eq_j}}(t)=\frac 1\mu\int_{t_0}^te^{\frac1\mu(t-\zeta)}u_{ISM_j}(\zeta)d\zeta$
$\mu \to$ Time constant of Filter
$\tilde u_{ISM_{eq_j}}(t_0)=0$

Integral term has to be redisgned as
$\phi_j(t)=-\sigma_j(t_0)-\int_{t_0}^tme_{2_j}(\zeta)+u_j(\zeta)+\tilde u_{ISM_{eq_j}}(\zeta)-u_{ISM_j}(\zeta)d\zeta$
$\phi_j(t_0)=\sigma_j(e(t_0))$
$\tilde u_{ISM_{eq_j}}=\hat h_j$

*Proposed controller is tested in the simulation on model of an industrial COMAU SMART3-S2 anthropomorphic robot manipulator identified on the basis of real data.*

## Conclusion
In this paper a switched structure control scheme, based on the combination of a decentralized control structure and a centralized inverse dynamics based control structure, has been proposed to solve a motion control problem for robot manipulators. A ISM control law is used in order to determine a switching rule to alternatively activate one of the two control structures. Apart from its robustness property, ISMallows one to estimate the matched and coupling uncertainty affecting the system, directly related to the required tracking performance of the robot.