11.4. Dynamic Equation of Motion (EoM)

The general form of robot dynamic equations (in joint space) is:

M(q)q¨+C(q,q˙)q˙ +G(q)=τ

Where: • M(q): Mass/Inertia matrix (n × n) • C(q,q˙): Coriolis and centrifugal force matrix • G(q): Gravity torque vector • τ: Vector of joint torques • q,q˙,q¨: Joint position, velocity, and acceleration vectors

The dynamic equation of motion for robots describes how the robot behaves under various forces and torques. This formula states that the sum of the forces acting on a robot, represented by the torque (τ), is equal to the combined effects of three main components:

1. Mass Matrix (M(q)): This matrix represents the robot's mass and how it changes depending on its configuration (positions of joints).
2. Coriolis and Centrifugal Forces (C(q, q˙)): This term accounts for the effects of rotation and changes in speed, which are important in multi-joint systems to ensure accurate motion prediction. It captures the non-linear interactions between joint velocities.
3. Gravity Torque Vector (G(q)): This vector accounts for the effects of gravity on the robot’s joints, ensuring the equations consider weight and stability. Together, these components define how different variables like joint position (q), joint velocity (q˙), and joint acceleration (q¨) influence the overall motion of the robot.

Think of the dynamic equations of motion as a recipe in a kitchen. Just like a recipe requires ingredients (in this case, M, C, and G) to create a dish (the robot's motion), these dynamic equations combine the forces acting on the robot to determine how it should move. If you change any ingredient in the recipe, the final dish (motion) will be different. For instance, if you add more flour (mass), it might affect how your cake rises (acceleration), just like changing a robot's mass can impact its motion.

• Mass Matrix M(q): Symmetric and positive definite. Depends on link masses and geometry. • Coriolis/Centrifugal Matrix C(q,q˙): Causes complex non-linear interactions in multi-DOF systems. • Gravity Vector G(q): Models the effect of gravity on each joint.

In this chunk, we break down each component of the dynamic equation:

1. Mass Matrix M(q): This matrix is crucial because it tells us how much 'weight' or 'inertia' each part of the robot has at different positions. It's symmetric, meaning its structure remains balanced, and it’s always positive, ensuring that our calculations remain valid. This matrix changes based on the robot's configuration, similar to how the layout of boxes might change whether they're stacked upright or lying down.
2. Coriolis/Centrifugal Matrix C(q, q˙): This matrix introduces complexity into our equations by accounting for the way joints affect each other in motion. In a multi-joint system, moving one joint can influence others due to their interconnected nature, leading to non-linear dynamics – for example, when driving a car, turning the steering wheel influences the direction of all the wheels, not just the front ones.
3. Gravity Vector G(q): This vector describes how gravity affects each joint of the robot. A robot on a slope will experience different gravitational effects than on level ground, resulting in variations in the torque needed to keep it stable.

Imagine you are carrying multiple bags of groceries (mass matrix) that shift in weight as you move (Coriolis). If you start walking uphill, you will have to exert more effort to carry them all due to gravity pulling downwards (gravity vector). Each of these aspects needs to be considered for a successful shopping trip, just as engineers consider each matrix's impact when programming a robot.

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