机器人学基础：力学与控制

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“Introduction_to_Robotics_Mechanics_and_Control -2.pdf” 是一本全英文的第三版机器人学教材，作者John J. Craig。本书主要关注机器人学中的机械操纵部分，涉及力学、控制理论和计算机科学等多个经典领域的基础知识。书中前8章探讨了机械工程和数学主题，中间3章覆盖了控制理论，最后两章则涉及计算机科学内容。本书适合高年级本科生或研究生学习，要求读者具备静态、动力学、线性代数的基础，以及一定的编程能力，有控制理论的先修知识会更有帮助。本书的一个核心目标是用简洁、直观的方式来讲解材料，不仅对机械工程师，也对电气工程师、计算机科学家和数学家等其他专业背景的学生友好。在斯坦福大学，这本书被广泛认为是一本易读且富有深度的教材。 “机器人运动学”和“运动控制”是本书的两个关键标签。运动学研究机器人的运动描述，包括位置、速度和加速度，而运动控制则涉及如何通过控制系统实现机器人精确、稳定的行为。在实际应用中，这两个概念对于设计和分析机器人的运动路径、执行任务以及与其他环境交互至关重要。书中可能包含如下具体内容： 1. **力学基础**：这部分可能涵盖牛顿力学、刚体动力学和多体系统分析，这些都是理解和设计机器人机构的基础。 2. **运动学**：包括笛卡尔运动学、迪卡尔运动逆解和雅可比矩阵，这些工具用于计算机器人关节速度和位置之间的关系。 3. **动力学**：讨论机器人动力学方程，如拉格朗日方程或牛顿-欧拉方法，用于分析力和扭矩如何影响机器人的运动。 4. **控制理论**：涵盖线性系统理论、状态空间表示、控制器设计（如PID控制器）以及稳定性分析，这些都是确保机器人按预定轨迹运动的关键。 5. **计算机科学**：可能涉及机器人程序设计、传感器数据处理和算法实现，可能使用高级编程语言进行示例和练习。 6. **实践应用**：书中可能包含案例研究和实验，帮助学生将理论知识应用于实际的机器人系统设计。通过学习这本书，读者将能够理解并应用基础的机器人学原理，为更深入的机器人技术研究和开发打下坚实基础。

8 Chapter 1

Introduction

FIGURE 1

9 A

World

War I biplane with a pilot and a rear gunner The rear gunner

mechanism is subject to

the

problem of singular positions.

enemy plane is

spotted at azimuth one o'clock and elevation 25 degrees!

The gunner trains his stream of bullets on the enemy plane and tracks its motion so

as to hit it with a continuous stream of bullets for as long as

possible. He succeeds

and thereby downs the enemy aircraft.

A second enemy plane is seen at azimuth one o'clock and elevation 70 degrees!

The gunner orients his gun and begins firing. The enemy plane is moving so as to

obtain a higher and higher elevation relative to the gunner's plane. Soon the enemy

plane is passing nearly overhead. What's this? The gunner is no longer able to keep

his stream of bullets trained on the enemy plane! He found that, as the enemy plane

flew overhead, he was required to change his azimuth at a very high rate. He was

not able to swing his gun in azimuth quickly enough, and the enemy plane

escaped!

In the latter scenario, the lucky enemy pilot was saved by a singularity!

The

gun's orienting mechanism, while working well over most of its operating range,

becomes less than ideal when the gun is directed straight upwards or nearly so. To

track targets that pass through the position directly overhead, a very fast motion

around the azimuth axis is required. The closer the target passes to the point directly

overhead, the faster the gunner must turn the azimuth axis to track the target. If

the target flies directly over the gunner's head, he would have to spin the gun on

its

azimuth axis at infinite speed!

Should the gunner complain to the mechanism designer about this problem?

Could a better mechanism be designed to avoid this problem? It turns out that

you really can't avoid the problem very easily. In fact, any

two-degree-of-freedom

orienting mechanism that has exactly two rotational joints cannot avoid having

this problem. In the case of this mechanism, with the stream of bullets directed

Section 1.2

The mechanics and control of mechanical manipulators

straight

up, their direction aligns with the axis of rotation of the azimuth rotation.

This means that, at exactly this point, the azimuth rotation does not cause a

change in the direction of the stream of bullets. We know we need two degrees

of freedom to orient the stream of bullets, but, at this point, we have lost the

effective use of one of the joints. Our mechanism has become locally degenerate

at this location and behaves as if it only has one degree of freedom (the elevation

direction).

This kind of phenomenon is caused by what is called a singularity of the

mechanism. All mechanisms are prone to these difficulties, including robots. Just

as with the rear gunner's mechanism, these singularity conditions do not prevent

a robot arm from positioning anywhere within its workspace. However, they can

cause problems with motions of the arm in their neighborhood.

Manipulators do not always move through space; sometimes they are also

required to touch a workpiece or work surface and apply a static force. In this

case the problem arises: Given a desired contact force and moment, what set of

joint torques is required to generate them? Once again, the Jacobian matrix of the

manipulator arises quite naturally in the solution of this problem.

Dynamics

Dynamics is a huge field of study devoted to studying the forces required to cause

motion. In order to accelerate a manipulator from rest, glide at a constant end-

effector velocity, and finally decelerate to a stop, a complex set of torque functions

must be applied by the joint actuators.2 The exact form of the required functions of

actuator torque depend on the spatial and temporal attributes of the path taken by

the end-effector and on the mass properties of the links and payload, friction in the

joints, and so on. One method of controlling a manipulator to follow a desired path

involves calculating these actuator torque functions by using the dynamic equations

of motion of the manipulator.

Many of us have experienced lifting an object that is actually much lighter

than we

(e.g., getting a container of milk from the refrigerator which

we thought was full, but was nearly empty). Such a misjudgment of payload can

cause an unusual lifting motion. This kind of observation indicates that the human

control system is more sophisticated than a purely kinematic scheme. Rather, our

manipulation control system makes use of knowledge of mass and other dynamic

effects. Likewise, algorithms that we construct to

the motions of a robot

manipulator should take dynamics into account.

A second use of the dynamic equations of motion is in simulation. By refor-

mulating the dynamic equations so that acceleration is computed as a function of

actuator torque, it is possible to simulate how a manipulator would move under

application of a set of actuator torques. (See Fig. 1.10.) As computing power

becomes more and more cost effective, the use of simulations is growing in use and

importance in many fields.

In Chapter 6, we develop dynamic equations of motion, which may be used to

control or simulate the motion of manipulators.

2We use joint actuators as the generic term for devices that

power a manipulator—for example,

electric motors, hydraulic and pneumatic actuators, and muscles.

Section 1.2

The mechanics and control of mechanical manipulators

FIGURE 1.1 1: In order to move the end-effector through space from point A

to point

B, we must compute a trajectory for each joint to follow.

a useful robot, that can take two approaches: build a specialized

robot for

a specific

task, or build a universal robot

that

would able to perform a wide variety of tasks.

In the case of a specialized robot, some careful thinking will yield a solution for

how many joints are needed. For example, a specialized robot designed solely to

place electronic components on a flat circuit board does not need to have

than four joints. Three joints allow the position of the hand to attain

any position

in three-dimensional space, with a fourth joint added to allow the hand to rotate

the grasped component about a vertical axis. In the case of a universal robot, it is

interesting that fundamental properties of the physical world

we live in dictate the

"correct" minimum number of joints—that minimum number is six.

Integral to the design of the manipulator are issues involving the choice and

location of actuators, transmission systems, and internal-position (and sometimes

force) sensors. (See Fig. 1.12.) These and other design issues will be discussed in

Chapter 8.

Linear

position control

Some

manipulators are equipped with stepper motors or other actuators that

can

execute a desired trajectory directly. However, the vast majority of manipulators

are driven by actuators that supply a force or a torque to cause motion of the links.

In this case, an algorithm is needed to compute torques that will cause the desired

motion. The problem of dynamics is central to the design of such algorithms, but

does not in itself constitute a solution. A primary concern of a position

control

system

is to compensate automatically for errors in knowledge of the parameters

of a system and to suppress disturbances that tend to perturb the system from the

desired trajectory. To accomplish this, position and velocity sensors

are monitored

by the control

algorithm,

which computes torque commands for the actuators. (See

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