11.5. Forward and Inverse Dynamics

Forward Dynamics

Given: • Joint torques τ Find: • Accelerations q¨ Solve: q¨ =M−1(τ −Cq˙ −G)

Forward dynamics is a process that calculates how a robot accelerates based on the joint torques applied to it. Here's how it works step-by-step:

1. Inputs: You start with the joint torques, denoted as τ. These are the forces applied at the joints of the robot.
2. Objective: The goal is to find the robot's accelerations (q¨), which indicate how fast the robot's positions will change over time.
3. Calculation: To make this calculation, you use a formula that combines the joint torques, the robot's mass and inertia properties (M), the Coriolis and centrifugal forces (Cq˙), and the effects of gravity (G). The equation looks like this: q¨ = M⁻¹(τ − Cq˙ − G). This means you calculate how the joint torques produce accelerations by adjusting for gravitational forces and other dynamic influences.
4. Result: The result gives you the robot's accelerations, allowing you to predict how it will move and respond to different forces.

This method is crucial in simulation environments to validate movements or to create trajectories for robot motion.

Imagine you're riding a bicycle. The pedaling force you exert on the pedals corresponds to the joint torques. The bike's acceleration depends on how hard you pedal (the torques) as well as the bike's weight and the slope of the hill (the dynamics). Just as you might calculate how fast you'll go downhill or uphill based on your pedaling and the bike's properties, forward dynamics calculates how a robot will accelerate given the forces on its joints.

Inverse Dynamics

Given: • Desired q,q˙,q¨ Find: • Required τ Used in: • Control systems (e.g., computed torque control) • Real-time torque computation

Inverse dynamics is the converse process of forward dynamics, where you determine the necessary joint torques to achieve a specific motion. Here’s how it unfolds:

1. Inputs: You start with desired states for the robot, which include positions (q), velocities (q˙), and accelerations (q¨). These are the outcomes you want the robot to achieve.
2. Objective: The goal now is to calculate the required joint torques (τ) that will cause the robot to reach these desired states.
3. Application: This calculation is essential in control systems, particularly in computed torque control, where the robot's system continuously adjusts its torque to maintain desired movements even as conditions change.
4. Outcome: The computed torques allow the robot to execute complex motions smoothly and responsively, ensuring accurate performance in real-time applications.

This method is key for enabling robots to interact dynamically with their environment.

Think of a basketball player trying to make a shot. The player knows the final position they want the ball to reach (the hoop) as well as how fast and how high the ball needs to travel. By understanding their arm's position and acceleration, they can figure out exactly how much force to apply (the torques) to make the shot successful. Similarly, in inverse dynamics, we determine the forces needed at the joints to achieve the desired motions of the robot.

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