Kinematics, Jacobian, Dynamics and Control - Planar RR Robot Example

1. Mass-Spring-Damper System
2. Mass-Spring-Damper Control
3. PID Control
4. Planar RR Robot
5. Forward Kinematics
6. Kinematic Jacobian
7. Numerical Inverse Kinematics
8. Dynamics, Euler-Lagrange equation, Moment of Inertia
9. Manipulator Control: PD+Gravity, Inverse Dynamics, and Feedforward
10. Geared robot dynamic model
11. References

Spring force comes from anything that stores mechanical energy when displaced:

• Actual coil spring: The most obvious case.
• Elastic deformation of materials: Beams, shafts, robot links, joints, seals, tires, belts, tendons, and even frames flex slightly and act like springs.
• Torsional stiffness in rotating systems: For rotation, the equivalent is This happens in flexible couplings, gear trains, harmonic drives, cable drives, etc.
• Gravity linearized around equilibrium: Near an equilibrium point, gravity can behave like an effective spring. Example: small-angle pendulum which looks like a rotational spring.
• Compressed fluids or air: Pneumatic systems, soft actuators, and trapped fluid volumes can create restoring forces.

Damping comes from anything that dissipates energy, usually into heat:

• Viscous friction: Oil dashpots, hydraulic dampers, lubricated sliding surfaces, fluid drag with .
• Material internal losses: Rubber, polymers, soft tissues, and metal structures under vibration lose energy internally. This is often called structural damping.
• Dry friction approximated as damping: Real Coulomb friction is not exactly proportional to velocity, but near some operating point engineers often approximate it as viscous damping.
• Air resistance / fluid drag: At low speeds this can be approximated as proportional to velocity.
• Electrical damping in motors: In electromechanical systems, back-EMF and electrical resistance can create damping-like behavior.
• Control damping: A feedback controller can intentionally create damping. Example: a PD controller adds an artificial spring term and damping term.

Characteristic equation:

Results in two distinct real roots: . Solution becomes . The system has no oscillation and return to equilibrium state slowly. Friction (responsible for damping) dominates stiffness.

Results in two complex conjugate roots: . Solution becomes , where . The system has oscillation with shrinking amplitude. Stiffness dominates damping in this case.

Results in equal real roots with . Solution becomes provides the fastest return to equilibrium without oscillation, balancing friction and stifness. This is the most desirable case in most control system.

Combining the two, with

How to choose the gains

• Set a desired closed-loop stiffness
• Obtain critical damping

Open loop equation (dynamics)

The control law consists of

• a model-based portion
  where
  Idea: If is taken as the new input, the system appears to be a unit mass.
• a servo portion
  Idea: Proceed as if the above were the open loop of a system to be controlled.

set for critical damping

Proportional-derivative (PD) controller:

The current controller maintains the block at .
How to enhance it to make the block track a prescribed trajectory?

desired trajectory:

tracking error:

Control law:

Controller:

With consideration of desired and

In the presence of a disturbance force , the error equation changes to:

Consider the simplest case of being a constant.

Set all derivatives in the error equation to zero (i.e., consider when the state stops changing):

Steady-State Error:

To eliminate the error, add an integral term to the control law.

Ziegler–Nichols method: is increased from zero to ultimate gain with stable output and consistent oscillations. and the oscillation period are then used to set the PID gains.

The second-order differential equation:

can be written in the standard state-space form:

Typical output choices are:

Root Locus Key Concepts
The root locus plot indicates how the closed loop poles of a system vary with a system parameter (typically a gain, K).

We can choose a value of 's' on this locus that will give us good results. The shape of the locus can also give us information on design of a more complex (lead/lag, PID controller).

Position of joint 2

The position of joint 2 (the end of the first link) is:

Position of the end-effector

The second link contributes a vector of length oriented at angle . Therefore,

So the forward kinematics are and the end-effector orientation is

For a planar 2R manipulator, the homogeneous transform of each link can be written as

From the last column, the end-effector position is again

The end-effector velocity is obtained by differentiating and .

Differentiate

Differentiate

By definition,

where the Jacobian is

Compute the partial derivatives:

Jacobian matrix:

Using the shorthand notation,

If you also want the end-effector orientation (), then

So the planar geometric Jacobian becomes

Thus,

For a revolute joint in planar motion, the linear velocity contribution is

where , and is the joint position.

• Joint 1 position:
• Joint 2 position:
• End-effector:

So the translational Jacobian is exactly the same as above.

These are useful for dynamics.
Let the COM distances from each joint be and .

Link 1 COM

Link 2 COM