Industrial Manipulators

An industrial manipulator of the type shown in Figure 1.1 is essentially a mechanical arm operating under computer control. Such devices, though far from the robots of science fiction, are nevertheless extremely complex electromechanical systems whose analytical description requires advanced methods, presenting many challenging and interesting research problems.

An official definition of such a robot comes from the Robot Institute of America (RIA):

A robot is a reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks.

The key element in the above definition is the reprogrammability, which gives a robot its utility and adaptability. The so-called robotics revolution is, in fact, part of the larger computer revolution.

Even this restricted definition of a robot has several features that make it attractive in an industrial environment. Among the advantages often cited in favor of the introduction of robots are decreased labor costs, increased precision and productivity, increased flexibility compared with specialized machines, and more humane working conditions as dull, repetitive, or hazardous jobs are performed by robots.

The industrial manipulator was born out of the marriage of two earlier technologies: teleoperators and numerically controlled milling machines. Teleoperators, or master-slave devices, were developed during the second world war to handle radioactive materials. Computer numerical control (CNC) was developed because of the high precision required in the machining of certain items, such as components of high performance aircraft. The first industrial robots essentially combined the mechanical linkages of the teleoperator with the autonomy and programmability of CNC machines.

The first successful applications of robot manipulators generally involved some sort of material transfer, such as injection molding or stamping, in which the robot merely attended a press to unload and either transfer or stack the finished parts. These first robots could be programmed to execute a sequence of movements, such as moving to a location A, closing a gripper, moving to a location B, etc., but had no external sensor capability. More complex applications, such as welding, grinding, deburring, and assembly, require not only more complex motion but also some form of external sensing such as vision, tactile, or force sensing, due to the increased interaction of the robot with its environment.

Figure 1.2 shows the estimated number of industrial robots worldwide between 2014 and 2020. In the future the market for service and medical robots will likely be even greater than the market for industrial robots. Service robots are defined as robots outside the manufacturing sector, such as robot vacuum cleaners, lawn mowers, window cleaners, delivery robots, etc. In 2018 alone, more than 30 million service robots were sold worldwide. The future market for robot assistants for elderly care and other medical robots will also be strong as populations continue to age.

Mobile Robots

Mobile robots encompass wheel and track driven robots, walking robots, climbing, swimming, crawling and flying robots. A typical wheeled mobile robot is shown in Figure 1.3. Mobile robots are used as household robots for vacuum cleaning and lawn mowing robots, as field robots for surveillance, search and rescue, environmental monitoring, forestry and agriculture, and other applications. Autonomous vehicles, for example self-driving cars and trucks, is an emerging area of robotics with great interest and promise.

There are many other applications of robotics in areas where the use of humans is impractical or undesirable. Among these are undersea and planetary exploration, satellite retrieval and repair, the defusing of explosive devices, and work in radioactive environments. Finally, prostheses, such as artificial limbs, are themselves robotic devices requiring methods of analysis and design similar to those of industrial manipulators.

The science of robotics has grown tremendously over the past twenty years, fueled by rapid advances in computer and sensor technology as well as theoretical advances in control and computer vision. In addition to the topics listed above, robotics encompasses several areas not covered in this text such as legged robots, flying and swimming robots, grasping, artificial intelligence, computer architectures, programming languages, and computer-aided design. In fact, the new subject of mechatronics has emerged over the past four decades and, in a sense, includes robotics as a subdiscipline. Mechatronics has been defined as the synergistic integration of mechanics, electronics, computer science, and control, and includes not only robotics, but many other areas such as automotive control systems.

Power Source

Most robots are either electrically, hydraulically, or pneumatically powered. Hydraulic actuators are unrivaled in their speed of response and torque producing capability. Therefore hydraulic robots are used primarily for lifting heavy loads. The drawbacks of hydraulic robots are that they tend to leak hydraulic fluid, require much more peripheral equipment (such as pumps, which require more maintenance), and they are noisy. Robots driven by DC or AC motors are increasingly popular since they are cheaper, cleaner and quieter. Pneumatic robots are inexpensive and simple but cannot be controlled precisely. As a result, pneumatic robots are limited in their range of applications and popularity.

Method of Control

Robots are classified by control method into servo and nonservo robots. The earliest robots were nonservo robots. These robots are essentially open-loop devices whose movements are limited to predetermined mechanical stops, and they are useful primarily for materials transfer. In fact, according to the definition given above, fixed stop robots hardly qualify as robots. Servo robots use closed-loop computer control to determine their motion and are thus capable of being truly multifunctional, reprogrammable devices.

Servo controlled robots are further classified according to the method that the controller uses to guide the end effector. The simplest type of robot in this class is the point-to-point robot. A point-to-point robot can be taught a discrete set of points but there is no control of the path of the end effector in between taught points. Such robots are usually taught a series of points with a teach pendant. The points are then stored and played back. Point-to-point robots are limited in their range of applications. With continuous path robots, on the other hand, the entire path of the end effector can be controlled. For example, the robot end effector can be taught to follow a straight line between two points or even to follow a contour such as a welding seam. In addition, the velocity and/or acceleration of the end effector can often be controlled. These are the most advanced robots and require the most sophisticated computer controllers and software development.

Application Area

Robot manipulators are often classified by application area into assembly and nonassembly robots. Assembly robots tend to be small and electrically driven with either revolute or SCARA geometries (described below). Typical nonassembly application areas are in welding, spray painting, material handling, and machine loading and unloading.

One of the primary differences between assembly and nonassembly applications is the increased level of precision required in assembly due to significant interaction with objects in the workspace. For example, an assembly task may require part insertion (the so-called peg-in-hole problem) or gear meshing. A slight mismatch between the parts can result in wedging and jamming, which can cause large interaction forces and failure of the task. As a result, assembly tasks are difficult to accomplish without special fixtures and jigs, or without controlling the interaction forces.

Geometry

Most industrial manipulators at the present time have six or fewer DOF. These manipulators are usually classified kinematically on the basis of the first three joints of the arm, with the wrist being described separately. The majority of these manipulators fall into one of five geometric types: articulated (RRR), spherical (RRP), SCARA (RRP), cylindrical (RPP), or Cartesian (PPP). We discuss each of these below in Section 1.3.

Each of these five manipulator arms is a serial link robot. A sixth distinct class of manipulators consists of the so-called parallel robot. In a parallel manipulator the links are arranged in a closed rather than open kinematic chain. Although we include a brief discussion of parallel robots in this chapter, their kinematics and dynamics are more difficult to derive than those of serial link robots and hence are usually treated only in more advanced texts.

Chapter 2: Rigid Motions

The first problem encountered is to describe both the position of the tool and the locations A and B (and most likely the entire surface S) with respect to a common coordinate system. In Chapter 2 we describe representations of coordinate systems and transformations among various coordinate systems. We describe several ways to represent rotations and rotational transformations and we introduce so-called homogeneous transformations, which combine position and orientation into a single matrix representation.

Chapter 3: Forward Kinematics

Typically, the manipulator will be able to sense its own position in some manner using internal sensors (position encoders located at joints 1 and 2) that can measure directly the joint angles θ₁ and θ₂. We also need therefore to express the positions A and B in terms of these joint angles. This leads to the forward kinematics problem studied in Chapter 3, which is to determine the position and orientation of the end effector or tool in terms of the joint variables.

It is customary to establish a fixed coordinate system, called the world or base frame to which all objects including the manipulator are referenced. In this case we establish the base coordinate frame o₀x₀y₀ at the base of the robot, as shown in Figure 1.19. The coordinates (x, y) of the tool are expressed in this coordinate frame as

(1.1)

(1.2)

in which a₁ and a₂ are the lengths of the two links, respectively. Also the orientation of the tool frame relative to the base frame is given by the direction cosines of the x₂ and y₂ axes relative to the x₀ and y₀ axes, that is,

(1.3)

which we may combine into a rotation matrix

(1.4)

Equations (1.1), (1.2), and (1.4) are called the forward kinematic equations for this arm. For a six-DOF robot these equations are quite complex and cannot be written down as easily as for the two-link manipulator. The general procedure that we discuss in Chapter 3 establishes coordinate frames at each joint and allows one to transform systematically among these frames using matrix transformations. The procedure that we use is referred to as the Denavit–Hartenberg convention. We then use homogeneous coordinates and homogeneous transformations, developed in Chapter 2, to simplify the transformation among coordinate frames.

Chapter 4: Velocity Kinematics

To follow a contour at constant velocity, or at any prescribed velocity, we must know the relationship between the tool velocity and the joint velocities. In this case we can differentiate Equations (1.1) and (1.2) to obtain

(1.5)

Using the vector notation and , we may write these equations as

(1.6)

The matrix J defined by Equation (1.6) is called the Jacobian of the manipulator and is a fundamental object to determine for any manipulator. In Chapter 4 we present a systematic procedure for deriving the manipulator Jacobian.

The determination of the joint velocities from the end-effector velocities is conceptually simple since the velocity relationship is linear. Thus, the joint velocities are found from the end-effector velocities via the inverse Jacobian

(1.7)

where J^{− 1} is given by

The determinant of the Jacobian in Equation (1.6) is equal to a₁a₂sin θ₂. Therefore, this Jacobian does not have an inverse when θ₂ = 0 or θ₂ = π, in which case the manipulator is said to be in a singular configuration, such as shown in Figure 1.20 for θ₂ = 0. The determination of such singular configurations is important for several reasons. At singular configurations there are infinitesimal motions that are unachievable; that is, the manipulator end effector cannot move in certain directions. In the above example the end effector cannot move in the positive x₂ direction when θ₂ = 0. Singular configurations are also related to the nonuniqueness of solutions of the inverse kinematics. For example, for a given end-effector position of the two-link planar manipulator, there are in general two possible solutions to the inverse kinematics. Note that a singular configuration separates these two solutions in the sense that the manipulator cannot go from one to the other without passing through a singularity. For many applications it is important to plan manipulator motions in such a way that singular configurations are avoided.

Chapter 5: Inverse Kinematics

Now, given the joint angles θ₁, θ₂ we can determine the end-effector coordinates x and y from Equations (1.1) and (1.2). In order to command the robot to move to location A we need the inverse; that is, we need to solve for the joint variables θ₁, θ₂ in terms of the x and y coordinates of A. This is the problem of inverse kinematics. Since the forward kinematic equations are nonlinear, a solution may not be easy to find, nor is there a unique solution in general. We can see in the case of a two-link planar mechanism that there may be no solution, for example if the given (x, y) coordinates are out of reach of the manipulator. If the given (x, y) coordinates are within the manipulator’s reach there may be two solutions as shown in Figure 1.21,

the so-called elbow up and elbow down configurations, or there may be exactly one solution if the manipulator must be fully extended to reach the point. There may even be an infinite number of solutions in some cases (Problem 1–19).

Consider the diagram of Figure 1.22. Using the law of cosines¹ we see that the angle θ₂ is given by

(1.8)

We could now determine θ₂ as θ₂ = cos ^{− 1}(D). However, a better way to find θ₂ is to notice that if cos (θ₂) is given by Equation (1.8), then sin (θ₂) is given as

(1.9)

and, hence, θ₂ can be found by

(1.10)

The advantage of this latter approach is that both the elbow-up and elbow-down solutions are recovered by choosing the negative and positive signs in Equation (1.10), respectively.

It is left as an exercise (Problem 1–17) to show that θ₁ is now given as

(1.11)

Notice that the angle θ₁ depends on θ₂. This makes sense physically since we would expect to require a different value for θ₁, depending on which solution is chosen for θ₂.

Chapter 6: Dynamics

In Chapter 6 we develop techniques based on Lagrangian dynamics for systematically deriving the equations of motion for serial-link manipulators. Deriving the dynamic equations of motion for robots is not a simple task due to the large number of degrees of freedom and the nonlinearities present in the system. We also discuss the so-called recursive Newton–Euler method for deriving the robot equations of motion. The Newton–Euler formulation is well-suited for real-time computation for both simulation and control applications.

Chapter 7: Path Planning and Trajectory Generation

The robot control problem is typically decomposed hierarchically into three tasks: path planning, trajectory generation, and trajectory tracking. The path planning problem, considered in Chapter 7, is to determine a path in task space (or configuration space) to move the robot to a goal position while avoiding collisions with objects in its workspace. These paths encode position and orientation information without timing considerations, that is, without considering velocities and accelerations along the planned paths. The trajectory generation problem, also considered in Chapter 7, is to generate reference trajectories that determine the time history of the manipulator along a given path or between initial and final configurations. These are typically given in joint space as polynomial functions of time. We discuss the most common polynomial interpolation schemes used to generate these trajectories.

Chapter 8: Independent Joint Control

Once reference trajectories for the robot are specified, it is the task of the control system to track them. In Chapter 8 we discuss the motion control problem. We treat the twin problems of tracking and disturbance rejection, which are to determine the control inputs necessary to follow, or track, a reference trajectory, while simultaneously rejecting disturbances due to unmodeled dynamic effects such as friction and noise. We first model the actuator and drive-train dynamics and discuss the design of independent joint control algorithms.

A block diagram of a single-input/single-output (SISO) feedback control system is shown in Figure 1.23. We detail the standard approaches to robot control based on both frequency domain and state space techniques. We also introduce the notion of feedforward control for tracking time-varying trajectories. We also introduce the fundamental notion of computed torque, which is a feedforward disturbance cancellation scheme.

Chapter 9: Nonlinear and Multivariable Control

In Chapter 9 we discuss more advanced control techniques based on the Lagrangian dynamic equations of motion derived in Chapter 6. We introduce the notion of inverse dynamics control as a means for compensating the complex nonlinear interaction forces among the links of the manipulator. Robust and adaptive control of manipulators are also introduced using the direct method of Lyapunov and so-called passivity-based control.

Chapter 10: Force Control

In the example robot task above, once the manipulator has reached location A, it must follow the contour S maintaining a constant force normal to the surface. Conceivably, knowing the location of the object and the shape of the contour, one could carry out this task using position control alone. This would be quite difficult to accomplish in practice, however. Since the manipulator itself possesses high rigidity, any errors in position due to uncertainty in the exact location of the surface or tool would give rise to extremely large forces at the end effector that could damage the tool, the surface, or the robot. A better approach is to measure the forces of interaction directly and use a force control scheme to accomplish the task. In Chapter 10 we discuss force control and compliance, along with common approaches to force control, namely hybrid control and impedance control.

Chapter 11: Vision-Based Control

Cameras have become reliable and relatively inexpensive sensors in many robotic applications. Unlike joint sensors, which give information about the internal configuration of the robot, cameras can be used not only to measure the position of the robot but also to locate objects in the robot’s workspace. In Chapter 11 we discuss the use of computer vision to determine position and orientation of objects.

In some cases, we may wish to control the motion of the manipulator relative to some target as the end effector moves through free space. Here, force control cannot be used. Instead, we can use computer vision to close the control loop around the vision sensor. This is the topic of Chapter 11. There are several approaches to vision-based control, but we will focus on the method of Image-Based Visual Servo (IBVS). With IBVS, an error measured in image coordinates is directly mapped to a control input that governs the motion of the camera. This method has become very popular in recent years, and it relies on mathematical development analogous to that given in Chapter 4.

Chapter 12: Feedback Linearization

Chapter 12 discusses the method of nonlinear feedback linearization. Feedback linearization relies on more advanced tools from differential geometry. We discuss the Frobenius theorem, from which we prove necessary and sufficient conditions for a single-input nonlinear system to be equivalent under coordinate transformation and nonlinear feedback to a linear system. We apply the feedback linearization method to the control of elastic-joint robots, for which previous methods such as inverse dynamics cannot be applied.

Chapter 13: Underactuated Systems

Chapter 13 treats the control problem for underactuated robots, which have fewer actuators than degrees of freedom. Unlike the control of fully-actuated manipulators considered in Chapters 8–11, underactuated robots cannot track arbitrary trajectories and, consequently, the control problems are more challenging for this class of robots than for fully-actuated robots. We discuss the method of partial feedback linearization and introduce the notion of zero dynamics, which plays an important role in understanding how to control underactuated robots.

Chapter 14: Mobile Robots

Chapter 14 is devoted to the control of mobile robots, which are examples of so-called nonholonomic systems. We discuss controllability, stabilizability, and tracking control for this class of systems. Nonholonomic systems require the development of new tools for analysis and control, not treated in the previous chapters. We introduce the notions of chain-form systems, and differential flatness, which provide methods to transform nonholonomic systems into forms amenable to control design. We discuss two important results, Chow’s theorem and Brockett’s theorem, that can be used to determine when a nonholonomic system is controllable or stabilizable, respectively.