Mechanisms

Joints and linkages

Joints constrain the motion between pairs of rigid bodies. For example, a revolute joint allows pure rotation of the bodies. A linear prismatic joint allows linear translation along an axis. There are some stylized symbols used to draw diagrams of linkages, collections of joints and rigid bodies (or links).

These symbols also allow some simple 3D mechanisms to be drawn in flat configurations without the need for a complicated perspective drawing.

The classification of linkages as parallel, branching, or serial in the figure depends on whether the arrangement of links and joints forms loops (also called cycles) or not. We will see later that computing the possible motions of parallel mechanisms can be difficult.

We use some notation to roughly describe the type of a serial robot arm. For example, an RR arm would indicate a serial arm with two revolute joints. A PR arm would indicate an arm with a prismatic joint and a revolute joint, in that order starting from the ground. We shorten this even further if there are multiple joints of the same type in a row — a 2R arm has two revolute joints.

The top two mechanisms in the figure above are a 2R and an RP arm.

Linkages as devices for computation

Linkages provide the transmission between the motors and the world in robots, but they can do more than that. For example, consider the problem of computing the value of sin(x). One approach would be to write a computer program that computes an an approximation using a series expansion. Another approach would be to build a linkage composed of an arm of length one connected by a revolute joint to another arm, like this:

The "input" for this computational machine is the joint angle, and the "output" is the length of the perpendicular from the arm of length one to the other arm. We have a name for this mechanism — it's a protractor! Slide rules are another example of a clever way to compute a function mechanically. I wonder if it's true that every function can be approximated arbitarily well by a planar mechanism composed of only rigid links, and revolute and prismatic joints?

The reason this idea is more than a curiosity is that clever linkage design can be one way of "programming" robot motion. A beautiful example of this is passive dynamic walking.

Workspaces for branching and serial mechanisms (graphical method)

Think about a robot arm bolted to the ground. The distal point on the arm is called the end effector, and is a natural place to put a tool of some type — fingers, a screw driver, or an arc welder. The set of points that the end effector can reach is called the robot workspace. Given a mechanism, what is its workspace? Here's an example with a planar 2R arm.

First, imagine all joints fixed (locked in place) except the last one. In our case, that means fixing the first joint. Move the last joint through its full range. In the general case, there might be limits on the range of motion of the joint, but in this case, the last joint can move freely; the end effector moves along the dashed circle shown in the left figure. This is the workspace of the last joint, assuming the other joints are fixed.

Now free the next joint closer to the ground, and move that joint through its full range of motion, dragging the workspace we've already constructed. In this case, the first joint drags the dashed circle in a big circle. The region that is swept out is a "donut", or annulus; this is the workspace for the arm.

Not all points in the workspace are the same. The figure on the right shows that there are two ways (or configurations of the arm) in which the end effector can reach any point on the interior of the workspace. We call these two configurations elbow up and elbow down configurations of the arm.

Each point on the outer boundary of the workspace can only be reached in one way (with the second joint fully extended, and the first joint at a particular angle), and each point on the inner boundary of the workspace can only be reached one way — with the second joint fully flexed.

This difference between the boundary of the workspace and the interior is interesting and important. Notice that for arm configurations in the interior, it is possible to move the effector instantaneously in any direction. However, for a point on the boundary, it is not possible to achieve an instantaneous velocity for the end effector that points inwards — first you have to bend the second joint, then you can move the effector inwards or outwards. We'll look at this phenomenon (called singularities of mechanisms) more closely when we study differential kinematics, but you might already be familiar with some uses of it: stiff-arming someone, standing straight to lift a heavy weight, or holding a suitcase with your arm at full extension.

Mobile robots

Here are examples of three common robot designs, from Acroname Robotics.

The first is a differential drive. There are two driving wheels. If both spin forwards at the same speed, the robot moves in a straight line forwards. If one spins in reverse, the robot spins in place. Different arcs of circles can be followed by choosing different differentials between the wheel speeds. Differential drives often have a third wheel, called a caster that is not driven and supports the weight of the robot. This wheel must be designed to allow slippage in every direction. You can find one common caster designon the bottom of office chairs — there's a wheel attached to a revolute joint, with the axis of the revolute joint offset so that it does not pass through the wheel axis. Another caster design is an omniwheel — a wheel with rollers along the rim that allow it to slip freely.

The second robot is a tricycle. The direction of travel is set by the angle of the steering wheel, while power is provided by the rear wheels. The tricycle can almost spin in place, but the configuration where the steering wheel is perpendicular to the other two is a singularity, and the robot in this configuration can only skid forwards or behave in an unpredictable way.

The third robot is a symmetric omnidirectional vehicle with three powered omniwheels. By choosing velocities for each of the three wheels appropriately, it is possible to achieve rotation and translation in any direction instantaneously.

We say that the tricycle and and the differential drive are underactuated — it is possible to move the vehicle sideways, but you cannot do so instantaneously (some sort of parallel parking maneuver will be needed). The omniwheeled robot is fully actuated. We'll look at this difference closely when we study differential kinematics.

Reuleaux collection of mechansims

There is a brilliant collection of different types of mechanisms that was built by Reuleaux in the 19th century. Cornell now has this collection and has a web page describing each mechanism:

Reuleaux mechanisms at Cornell

You should take a look!