Chapter 4: Kinematics and Dynamics of Robots

# **Chapter 4: Kinematics and Dynamics of Robots**  

Abstract 

Robot kinematics and dynamics are the study of how a robot moves, and the forces that cause that motion. Both are fundamental to robotics and are used to design and control robots. 

Kinematics 

Analyzes motion without considering forces
Uses geometric properties like lengths and degrees of freedom
Used for motion planning
Can be used to describe the position, orientation, and motion of a robot's joints

Dynamics Analyzes motion caused by forces, Uses parameters like mass and inertia, Used for motion control, and Represents the relationship between joint torques and robot motion. 

How kinematics and dynamics are related 

Kinematics and dynamics are closely related and complement each other
Kinematics is usually easier to solve than dynamics
Kinematics can have multiple solutions, while dynamics has a unique solution

Applications 

• Kinematics and dynamics models are used in the simulation of motion, analysis of robot manipulator structures, and design of control algorithms

So let's explore the Chapter 4 in detail

## **4.1 Introduction**  

Robots are mechanical systems designed to perform tasks autonomously or semi-autonomously. Understanding their motion requires studying **kinematics**, which deals with motion without considering forces, and **dynamics**, which considers the forces causing the motion. This chapter explores these concepts, including forward and inverse kinematics, velocity analysis, Jacobian matrices, and dynamic modeling of robots.  

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## **4.2 Kinematics of Robots**  

Kinematics describes the motion of a robot in terms of position, velocity, and acceleration without considering forces. It is divided into **forward kinematics** and **inverse kinematics**.  

### **4.2.1 Forward Kinematics (FK)**  

Forward kinematics determines the position and orientation of the robot’s end-effector (tool or gripper) based on given joint parameters.  

#### **Mathematical Formulation**  

For an \(n\)-joint robotic manipulator, let:  

- \( q = [q_1, q_2, ..., q_n]^T \) be the joint variables (angles for revolute joints, displacements for prismatic joints).  

- \( T_0^n \) be the transformation matrix that represents the end-effector position and orientation relative to the base.  

Using **Denavit-Hartenberg (D-H) parameters**, the homogeneous transformation matrix for each joint is:  

\[

T_i^{i+1} =

\begin{bmatrix}

\cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i \cos\theta_i \\

\sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i \sin\theta_i \\

0 & \sin\alpha_i & \cos\alpha_i & d_i \\

0 & 0 & 0 & 1

\end{bmatrix}

\]

where:  

- \( \theta_i \) = joint angle  

- \( d_i \) = link offset  

- \( a_i \) = link length  

- \( \alpha_i \) = link twist angle  

By multiplying individual transformations, we obtain:  

\[

T_0^n = T_0^1 T_1^2 \dots T_{n-1}^n

\]

This equation provides the **pose** (position and orientation) of the end-effector.  

### **4.2.2 Inverse Kinematics (IK)**  

Inverse kinematics calculates the required joint values to reach a given end-effector position and orientation. Unlike FK, which has a unique solution, IK may have multiple or no solutions.  

#### **Analytical and Numerical Approaches**  

1. **Analytical Approach:**  

   - Closed-form solutions using trigonometry and algebra.  

   - Works well for simple robots (e.g., 2-DOF or 3-DOF manipulators).  

2. **Numerical Approach:**  

   - Iterative methods such as **Newton-Raphson** and **Jacobian Inverse**.  

   - Used for high-DOF robots with complex kinematics.  

The general equation for IK is:  

\[

q = f^{-1}(X)

\]

where \( X \) is the desired end-effector pose, and \( q \) is the joint configuration.  

---

## **4.3 Velocity and Jacobian Analysis**  

Velocity kinematics describes the relationship between joint velocities and end-effector velocities.  

### **4.3.1 Jacobian Matrix**  

The **Jacobian matrix** \( J \) maps joint velocities \( \dot{q} \) to end-effector velocities \( \dot{X} \):  

\[

\dot{X} = J(q) \dot{q}

\]

where:  

\[

J(q) =

\begin{bmatrix}

\frac{\partial x}{\partial q_1} & \frac{\partial x}{\partial q_2} & ... & \frac{\partial x}{\partial q_n} \\

\frac{\partial y}{\partial q_1} & \frac{\partial y}{\partial q_2} & ... & \frac{\partial y}{\partial q_n} \\

\frac{\partial z}{\partial q_1} & \frac{\partial z}{\partial q_2} & ... & \frac{\partial z}{\partial q_n}

\end{bmatrix}

\]

**Singularities:**  

- When \( \det(J) = 0 \), the Jacobian is singular, leading to loss of motion control.  

**Inverse Kinematics using Jacobian:**  

The inverse kinematics problem can be solved iteratively using:  

\[

\dot{q} = J^{-1} \dot{X}

\]

If \( J \) is non-square, the **pseudo-inverse** is used:  

\[

\dot{q} = J^T (JJ^T)^{-1} \dot{X}

\]

---

## **4.4 Dynamics of Robots**  

Robot dynamics studies the forces and torques required to achieve desired motion. It is governed by Newton-Euler and Lagrange formulations.  

### **4.4.1 Newton-Euler Equations**  

Newton’s laws are applied to each link of the robot:  

\[

F = m \ddot{x}

\]

\[

\sum \tau = I \ddot{\theta}

\]

where:  

- \( F \) = force  

- \( \tau \) = torque  

- \( m \) = mass of link  

- \( I \) = moment of inertia  

The recursive Newton-Euler algorithm computes forces and torques efficiently.  

### **4.4.2 Lagrange Formulation**  

Lagrangian dynamics use **energy-based** methods to derive equations of motion.  

The **Lagrange equation** is:  

\[

\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = \tau_i

\]

where:  

- \( L = K - P \) is the **Lagrangian function** (Kinetic - Potential Energy).  

- \( \tau_i \) is the applied torque on joint \( i \).  

Using this, the **equations of motion** for an \( n \)-DOF manipulator are:  

\[

M(q) \ddot{q} + C(q, \dot{q}) \dot{q} + G(q) = \tau

\]

where:  

- \( M(q) \) = Mass/Inertia matrix  

- \( C(q, \dot{q}) \) = Coriolis and centrifugal forces  

- \( G(q) \) = Gravity forces  

---

## **4.5 Motion Planning and Control**  

Once kinematics and dynamics are understood, motion planning ensures smooth, collision-free paths for the robot.  

### **4.5.1 Trajectory Planning**  

A robot's trajectory includes positions, velocities, and accelerations over time. Common methods:  

- **Polynomial Interpolation** (Cubic, Quintic Splines)  

- **Trapezoidal and S-Curve Velocity Profiles**  

### **4.5.2 Robot Control**  

Robots require control strategies to execute motions accurately:  

1. **PID Control:** Proportional-Integral-Derivative control for precise positioning.  

2. **Computed Torque Control:** Uses dynamic equations for better control.  

3. **Adaptive and Model Predictive Control:** Adjusts to changing conditions.  

---

## **4.6 Conclusion**  

This chapter covered fundamental concepts of robot kinematics and dynamics. Kinematics helps determine motion, while dynamics accounts for forces. Understanding these principles enables advanced applications in robotic control, motion planning, and real-world automation.  

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## **4.7 References**  

1. Craig, J. J. (2005). *Introduction to Robotics: Mechanics and Control.* Pearson.  

2. Spong, M. W., Hutchinson, S., & Vidyasagar, M. (2020). *Robot Modeling and Control.* Wiley.  

3. Siciliano, B., & Khatib, O. (2016). *Springer Handbook of Robotics.* Springer.