3.6 Manipulators and grippers

Types of manipulators

Manipulators are the parts of a robot that let it physically interact with its environment: picking things up, moving them, assembling parts, or using tools. Choosing the right manipulator depends on the application, the workspace the robot needs to reach, how much weight it needs to carry, and how many degrees of freedom it requires.

Manipulators are classified based on their kinematic structure, joint types, and how they move through space.

Serial vs parallel manipulators

These are the two fundamental structural categories:

• Serial manipulators connect links and joints in an open chain, one after another, like a typical robotic arm. They offer a large workspace and high dexterity, but they tend to have lower stiffness and payload capacity because forces accumulate along the chain.
• Parallel manipulators use multiple kinematic chains connecting the base to the end effector simultaneously, forming closed loops. The Stewart platform (used in flight simulators) is a classic example. They provide higher stiffness, accuracy, and payload capacity, but their workspace is more limited and they're less dexterous.

Think of it this way: serial is like your arm (one chain of joints), while parallel is like a set of pistons all pushing on the same platform from different angles.

Cartesian manipulators

Cartesian manipulators use three prismatic joints (joints that slide rather than rotate) aligned with the x, y, and z axes. Because the joint axes map directly onto Cartesian coordinates, control is straightforward. You command motion along each axis independently.

• Workspace: a rectangular box defined by each joint's travel range
• Common applications: 3D printers, CNC machines, gantry systems
• Strength: precise linear movements with simple control
• Limitation: bulky structure relative to workspace size

Cylindrical manipulators

Cylindrical manipulators combine one revolute joint (rotation about the base) with two prismatic joints (radial and vertical linear motion). The workspace forms a cylindrical volume.

• The revolute joint rotates the arm around the base axis
• One prismatic joint extends/retracts radially, the other moves vertically
• Common applications: material handling, assembly tasks that need rotation plus reach

Spherical manipulators

Also called polar manipulators, these have two revolute joints and one prismatic joint. The revolute joints handle rotation about the base and elevation, while the prismatic joint extends radially.

• Workspace: a partial spherical volume
• Common applications: spray painting, welding, tasks needing wide angular range but limited linear reach

SCARA manipulators

SCARA stands for Selective Compliance Assembly Robot Arm. It has two revolute joints that rotate in the horizontal plane and one prismatic joint for vertical motion.

• Very fast and precise for horizontal movements
• Workspace: a cylindrical volume with limited vertical range
• Common applications: pick-and-place operations, electronics assembly, packaging
• The "selective compliance" means the arm is rigid vertically but compliant horizontally, which is ideal for insertion tasks

Articulated manipulators

Articulated (or anthropomorphic) manipulators use a series of revolute joints, mimicking the structure of a human arm. They typically have six degrees of freedom: three joints for positioning and three for orienting the end effector.

• Workspace: a complex volume determined by joint limits and link lengths
• Highest flexibility and versatility among manipulator types
• Common applications: welding, painting, material handling, general-purpose industrial robots
• The trade-off is more complex control due to the nonlinear relationships between joints

Manipulator kinematics

Kinematics studies motion without worrying about the forces causing it. For manipulators, this means figuring out the geometric relationship between joint angles and where the end effector ends up. You need kinematics to control the robot's movements, plan trajectories, and solve for joint configurations.

There are two core problems: forward kinematics and inverse kinematics.

Forward kinematics

Forward kinematics answers: given the current joint angles, where is the end effector?

This is the easier problem. You plug joint angles into the kinematic equations, and out comes the end effector's position and orientation. The equations are derived from the manipulator's geometry using the Denavit-Hartenberg (DH) convention.

Forward kinematics is essential for visualizing the robot's current configuration and predicting where the end effector will go as joints move.

Inverse kinematics

Inverse kinematics answers the opposite question: given a desired end effector position and orientation, what joint angles do you need?

This is significantly harder for several reasons:

• There may be multiple valid solutions (the robot can reach the same point in different configurations)
• There may be no solution (the target is outside the workspace)
• The equations are often nonlinear and coupled

Two main approaches exist for solving inverse kinematics:

• Analytical methods derive closed-form equations, giving exact solutions when they exist
• Numerical methods use iterative algorithms to converge on a solution, which is more general but slower

Denavit-Hartenberg parameters

The Denavit-Hartenberg (DH) convention provides a systematic way to assign coordinate frames to each link of a manipulator. It uses four parameters per joint to describe the geometric relationship between adjacent frames:

• $a$ (link length): distance between the z-axes of adjacent frames
• $\alpha$ (link twist): angle between the z-axes of adjacent frames
• $d$ (link offset): distance between the x-axes of adjacent frames
• $\theta$ (joint angle): angle between the x-axes of adjacent frames

For a revolute joint, $\theta$ is the variable. For a prismatic joint, $d$ is the variable. The other parameters are fixed constants determined by the robot's physical geometry.

The DH convention simplifies deriving kinematic equations because each joint's transformation can be expressed as a standard 4×4 homogeneous transformation matrix, and you multiply them together to get the full forward kinematics.

Jacobian matrix

The Jacobian matrix connects joint velocities to end effector velocities (both linear and angular). It's a matrix of partial derivatives that captures the instantaneous relationship between joint space and Cartesian space.

The Jacobian is used for:

• Velocity control: commanding end effector velocity by computing the required joint velocities
• Singularity analysis: finding configurations where the manipulator loses one or more degrees of freedom (the Jacobian loses rank), which causes problems like infinite joint velocities
• Manipulability analysis: evaluating how well the manipulator can move in different directions from a given configuration

The Jacobian is computed from the kinematic equations using the chain rule of differentiation. Near singularities, the Jacobian becomes ill-conditioned, so the robot struggles to move in certain directions.

Manipulator dynamics

Dynamics goes beyond kinematics by accounting for the forces and torques that cause motion. You need dynamics to design controllers, optimize trajectories, and ensure the robot operates safely and efficiently.

Two main formulations exist for deriving the equations of motion.

Lagrangian formulation

The Lagrangian approach is energy-based. You compute the system's kinetic energy $T$ and potential energy $V$, then define the Lagrangian as:

$L = T - V$

Applying the Euler-Lagrange equations to $L$ produces a set of second-order differential equations describing the manipulator's dynamics. This formulation yields compact, systematic equations that are well-suited for control design and theoretical analysis.

Newton-Euler formulation

The Newton-Euler approach is force-based. It applies Newton's second law and Euler's rotational equations to each link, then recursively propagates forces and torques from the base outward to the end effector (and back).

This produces coupled first-order differential equations. The Newton-Euler formulation is more intuitive physically (you can see the forces on each link) and is computationally efficient, making it preferred for real-time control implementations.

Equations of motion

Both formulations lead to the same general form:

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

Where:

• $M(q)$: inertia matrix, depends on joint positions. Captures how much torque is needed to accelerate each joint.
• $C(q,\dot{q})$: Coriolis and centrifugal matrix, depends on joint positions and velocities. Accounts for velocity-dependent coupling forces between joints.
• $G(q)$: gravity vector, depends on joint positions. Represents the torques needed to counteract gravity.
• $\tau$: joint torque vector, the control inputs you apply.

These equations are the foundation for model-based control techniques like computed torque control and model predictive control.

Dynamic model simplification

The full dynamic model can be computationally expensive, especially for robots with many degrees of freedom. Common simplifications include:

• Neglecting Coriolis and centrifugal terms when the robot moves slowly (these forces scale with velocity squared)
• Assuming a diagonal inertia matrix, which treats each joint as dynamically independent of the others
• Using reduced-order models that capture only the dominant dynamics

These simplifications trade accuracy for speed, which matters for real-time control and simulation. The key is knowing when the simplification is valid for your application.

Manipulator control

Control algorithms make the manipulator follow desired trajectories, exert desired forces, or complete specific tasks. Because manipulator dynamics are nonlinear and coupled, control design is more involved than for simple linear systems.

Joint space vs Cartesian space control

• Joint space control works directly with joint variables (angles or displacements). It's simpler to implement because each joint can be controlled somewhat independently. However, you need inverse kinematics to convert a desired end effector pose into joint commands.
• Cartesian space control (also called operational space control) works directly with the end effector's position and orientation. This is more intuitive for task specification but requires computing the Jacobian matrix in real time and handling redundancy if the robot has more than six degrees of freedom.

PID control

PID (Proportional-Integral-Derivative) control is the most common feedback controller for manipulators, especially at the individual joint level.

• Proportional (P): produces a control signal proportional to the current error. Larger error means stronger correction.
• Integral (I): accumulates past errors over time and eliminates steady-state offset. Without it, the joint might settle slightly off target.
• Derivative (D): responds to the rate of change of error, damping oscillations and improving transient response.

PID is simple to implement and tune, but it treats each joint independently. For highly nonlinear, coupled manipulators, PID alone may not provide optimal tracking performance.

Computed torque control

Computed torque control (also called inverse dynamics control) uses the full dynamic model to cancel out nonlinear terms. The idea is:

1. Use the dynamic model to compute what torques the nonlinear dynamics would produce
2. Apply feedforward torques that cancel those dynamics out
3. Add a feedback term (often PD) to stabilize the system and track the desired trajectory

The result is a linearized, decoupled system that's much easier to control. The catch: you need an accurate dynamic model. If the model is wrong, the cancellation is imperfect and performance degrades.

Adaptive control

Adaptive control adjusts controller parameters in real time to handle uncertainties or changes in the system. This is useful when:

• The manipulator's dynamic parameters aren't precisely known
• The payload changes (picking up objects of different mass)
• Wear and tear alter the system over time

Adaptive algorithms estimate unknown parameters online and update controller gains accordingly. Common approaches include model reference adaptive control (MRAC) and self-tuning regulators (STR).

Force control

Many tasks require controlling not just position but the forces the robot exerts on its environment (assembly, grinding, polishing, surface finishing). Pure position control can generate dangerous forces if the environment is stiff.

Two main approaches:

• Hybrid position/force control: divides the task space into directions where you control position and directions where you control force. For example, when polishing a surface, you control force normal to the surface and position along it.
• Impedance control: instead of controlling force directly, you control the dynamic relationship between motion and force. The robot behaves like a mass-spring-damper system, allowing it to comply with the environment naturally.

Both approaches require force/torque sensors (or force estimation techniques) to measure interaction forces.

End effectors and grippers

End effectors are whatever is attached to the tip of the manipulator. Grippers are end effectors specifically designed for grasping and manipulating objects. The right gripper depends on the object's shape, size, weight, surface properties, and what the task demands.

Mechanical grippers

Mechanical grippers use physical elements like jaws, fingers, or claws to grasp objects. They're actuated by electric motors, pneumatic cylinders, or hydraulic systems.

• Parallel jaw grippers: two flat jaws move toward each other in parallel. Simple, reliable, good for prismatic objects.
• Angular jaw grippers: jaws pivot rather than translate, useful for different object geometries.
• Three-finger grippers: offer more stable grasps on irregular shapes by providing three contact points.

Mechanical grippers are the most versatile type and can handle a wide range of object shapes and sizes by swapping jaw configurations.

Vacuum grippers

Vacuum grippers use suction cups to create a pressure difference that holds the object. They work best on flat, smooth, non-porous surfaces like glass, plastic sheets, or metal panels.

• Fast attachment and release
• Gentle on delicate surfaces
• Require a vacuum generator and pressure regulation
• Won't work on porous, rough, or highly curved surfaces

Magnetic grippers

Magnetic grippers attract and hold ferromagnetic objects (iron, steel, nickel) using either electromagnets or permanent magnets.

• Electromagnets can be switched on and off, making pickup and release controllable
• Permanent magnets provide constant force but need a mechanical release mechanism
• Simple, robust, and work in harsh environments
• Limited to ferromagnetic materials only

Adhesive grippers

Adhesive grippers create a temporary bond with the object's surface. The adhesion can be based on different principles:

• Electrostatic adhesion: uses electric charge to attract objects
• Van der Waals forces: molecular-level attraction, similar to how gecko feet work
• Gecko-inspired microstructures: synthetic surfaces with tiny features that mimic gecko toe pads

These grippers handle lightweight, delicate, or non-magnetic objects well. They need a release mechanism and a way to control adhesion strength.

Soft grippers

Soft grippers are made from compliant, deformable materials (silicone, rubber, elastomers) that conform to the object's shape during grasping. They can be actuated by:

• Pneumatic or hydraulic pressure (inflating chambers to curl fingers)
• Shape memory alloys
• Other soft actuation methods

Soft grippers distribute grasping force evenly across the contact surface, reducing the risk of damage. They're ideal for handling delicate items (fruit, biological tissue), irregularly shaped objects, or fragile components. The trade-off is lower precision and payload capacity compared to rigid grippers.

Gripper design considerations

Designing a gripper requires balancing multiple factors to match the task requirements. A well-designed gripper securely grasps objects without damaging them, fits within the manipulator's payload limits, and works reliably across the expected range of objects.

Payload capacity

Payload capacity is the maximum weight the gripper can handle safely. It depends on the gripper's size, material strength, actuation mechanism, and the manipulator's own payload limit.

Always include a safety factor to account for dynamic loads (acceleration forces during fast movements) and uncertainties (object weight variation). Overloading risks deformation or dropped objects. Over-designing wastes weight and cost.

Gripping force

Gripping force is how hard the gripper squeezes the object. It must be:

• High enough to prevent slippage during acceleration, deceleration, and external disturbances
• Low enough to avoid crushing or deforming the object

The required gripping force depends on the object's weight, the coefficient of friction between the jaw and the object, and the dynamic forces during motion. For fragile objects, force sensors or compliance in the gripper help prevent damage.

Jaw shape and material

The jaw geometry and material directly affect grip quality:

• Jaw shapes can be flat (for prismatic objects), curved (for cylindrical objects), or custom-contoured to match specific parts. Better contact geometry means more even force distribution and a more stable grasp.
• Jaw materials are chosen based on friction requirements, wear resistance, chemical compatibility, and whether the object can be scratched or marked. Common choices include metal alloys (durable, high friction), polymers (lightweight, chemically resistant), elastomers (high friction, gentle on surfaces), and composites.