Forward/inverse kinematics and transformation matrices

Question

I have found the homogeneous transformation matrix that can be used to determine the relation between the parent and child links of a robot. This matrix is shown below.

$T_i = \left[\begin{array}{ c c c c } \cos \theta_{i} & -\sin \theta_{i} & 0 & a_{i-1} \\ \sin \theta_{i} \cos \alpha_{i-1} & \cos \theta_{i} \cos \alpha_{i-1} & -\sin \alpha_{i-1} & -\sin \alpha_{i-1} d_{i} \\ \sin \theta_{i} \sin \alpha_{i-1} & \cos \theta_{i} \sin \alpha_{i-1} & \cos \alpha_{i-1} & \cos \alpha_{i-1} d_{i} \\ 0 & 0 & 0 & 1 \end{array}\right]$

I have used this matrix together with the Denavit Hartenberg parameters to make a Matlab Rigidbody Tree model. All the joints of the robot are behaving correctly. Does this mean that these matrices solve the forward kinematics? And what steps would I have to take to get the inverse kinematics in a Simulink block for example?

Answer 1

For each joint relation you model, you would get one homogeneous transformation matrix (HTM) T $$T^{i-1}_i$$ If you multiply all HTM from base to end-effector, you would get what is called forward kinematic matrix FK $$FK(\theta) = T^0_N= T^0_1*T^1_2*...*T^{N-1}_N$$ which map joint configuration into cartesian space (represent pose of end-effector). Then for inverse kinematic there are two main method, graphical method and analytical method. For low number DOF (degree of freedom) (DOF<3), you can perform graphical method, by calculating end-effector and joint relation using simple trigonometry function. For high number DOF (DOF>3) , you should perform analytical method. This method really complex, but luckily there are several library available for MatLab which you can use such as RTB or MatLab official library.

There is one indirect method of inverse kinematic, that is by using Jacobian matrix.