Robot manipulator design, part 2: advanced topics

Part 1 covered degrees of freedom, joint types and arm layouts. This part goes one step further: where can the arm reach, where does it get into trouble, how do we measure how good it is, and how do we control it?

Three questions before you start

What task will the robot do? Lifting, welding, assembling or sorting. The task decides the structure and the number of joints.
Where will it work? The size and shape of the space it must reach, and what else is in that space.
What will it carry? The payload drives material choice, joint design and motor size. Strong, but not heavier than it needs to be.

Robot arm and human arm

Human	Robot	Role
Torso	Base	Fixed reference
Shoulder, elbow	First three joints	Position the wrist
Wrist	Last three joints	Orient the tool
Hand	End effector	Grip or work on the object

Workspace analysis

The workspace is every point the tool can reach. Engineers distinguish two kinds:

Reachable workspace: points the tip can reach in at least one orientation.
Dexterous workspace: points it can reach in every orientation. It is always smaller.

For a two-link planar arm, the reachable workspace is a ring around the base:

$$ |l_1 - l_2| \;\le\; \sqrt{x^2 + y^2} \;\le\; l_1 + l_2 $$

Example. With $l_1 = 1.0$ m and $l_2 = 0.8$ m the arm reaches everything between 0.2 m and 1.8 m from the base, as long as joint limits do not cut parts away.

Two ways to find the workspace of a real arm, and it is good practice to use both:

Graphical: model the arm in CAD and sweep each joint through its range.
Analytical / numerical: compute the forward kinematics for many joint combinations and plot the tip positions.

If the two agree, your model is right.

Singularities

A singularity is a pose where the arm loses a direction of motion. Mathematically the Jacobian $\mathbf{J}$ loses rank, and $\det\mathbf{J} = 0$ for a square Jacobian. Near a singularity, small tool motions need very large joint speeds, which is dangerous.

Type	Where	Example
Boundary	At the edge of the workspace	Arm fully stretched out
Interior	Inside the workspace, when two joint axes line up	Wrist singularity of a 6-axis arm, when axes 4 and 6 align

Planners avoid them by keeping a safety distance, or by slowing down when the arm gets close.

Redundancy

An arm with more joints than the task needs is redundant. A 7-DOF arm doing a 6-DOF task has one spare motion, a self-motion: the elbow can move while the tool stays still. That spare freedom can be used to avoid obstacles, stay away from singularities and joint limits, or reduce energy. Redundant arms are solved with the pseudo-inverse of the Jacobian plus a term that moves in the "spare" direction:

$$ \dot{\mathbf{q}} = \mathbf{J}^{+}\dot{\mathbf{x}} + \left(\mathbf{I} - \mathbf{J}^{+}\mathbf{J}\right)\dot{\mathbf{q}}_0 $$

The second term does not move the tool at all; it only uses the extra freedom to pursue the secondary goal $\dot{\mathbf{q}}_0$.

Measuring performance

Measure	Meaning
Payload	Heaviest load it can carry at full speed
Reach	Largest distance from the base to the tool
Repeatability	How closely it returns to the same taught point (often ±0.02 to ±0.1 mm in industry)
Accuracy	How close it gets to a computed point; usually worse than repeatability
Speed and cycle time	How fast it completes a standard motion
Manipulability	How freely it can move in every direction at a given pose

Manipulability (Yoshikawa) turns the last row into a number:

$$ w = \sqrt{\det\!\left(\mathbf{J}\mathbf{J}^\top\right)} $$

It is zero at a singularity and large where the arm moves easily in all directions. For the two-link arm it is simply $w = l_1 l_2 |\sin\theta_2|$, largest with the elbow at 90°.

Control strategies

From simplest to most advanced:

Strategy	Idea	Good for
Independent joint PID	Each joint has its own PID loop	Slow arms, high gear ratios
Computed torque	Use the dynamic model to cancel gravity and inertia	Fast, accurate motion
Impedance / force control	Control how stiff the arm feels, not only where it is	Contact tasks, working next to people
Visual servoing	Close the loop through a camera	Grasping objects whose position varies

Computed-torque control uses the arm's dynamic model so that each joint behaves like a simple, well-tuned system:

$$ \boldsymbol{\tau} = \mathbf{M}(\mathbf{q})\left(\ddot{\mathbf{q}}_d + \mathbf{K}_d\,\dot{\mathbf{e}} + \mathbf{K}_p\,\mathbf{e}\right) + \mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\,\dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}), \qquad \mathbf{e} = \mathbf{q}_d - \mathbf{q} $$

Why it is worth the effort

Well-designed robot arms save time, cut costs, take over dangerous or repetitive work, reduce errors and run around the clock. Getting the workspace, singularities and control right from the start is what makes that possible.

Watch the video

For the full explanation, see Robot manipulator design, part 2 on YouTube.