Spong Published Computer Science From the Publisher: This self-contained introduction to practical robot kinematics and dynamics includes a comprehensive treatment of robot control.
Provides background material on terminology and linear transformations, followed by coverage of kinematics and inverse kinematics, dynamics, manipulator control, robust control, force control, use of feedback in nonlinear systems, and adaptive control. Each topic is supported by examples of specific applications.
Derivations and proofs are… Expand. Save to Library Save. Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations. Figures, Tables, and Topics from this paper. Inverse kinematics Robust control Robot control Nonlinear system. Citation Type. Has PDF. Publication Type. More Filters. Dynamics and control of direct-drive robots with positive joint torque feedback.
Engineering, Computer Science. We also introduce the nutiun of feedforward control and the techniques of computed torque and inverse dynamics as a means tor compensating the complex nonlinear interaction forces among the links of the mani- pulator. Chapters Ten and Eleven provide some ad- ditional advanced techniques from nonlinear control tbeory tbat arc useful for controlling high performance robots.
This would be: quite difficult to accomplish in practice, howevl. MA, Louis, EN, Y. Freeman, San Frand! Winston, New York, 19R5. For each application di. I-H List fivc 'lpplieations where computer vision would he useful in robotics. What would be some of the economic and.!
What would be some of the c ,;onomic and social conseqUcllCC:' of such an act? A linear axis would travel the distance d while a rotational link would travel through an arclength f a as shown. I-tS A single-link n:volute ann is shown in Figure Assume perfect gears. Plot the time history of joint angles. If so, explain how this clin occur. Indeed, the geometry of three-dimensional space and of rigid mo- tfons plays a central role in all aspects of robotic manipulation.
In this chapter we study the operations of rotation and translation and intro- duce the notion of homogeneous transformations. We also investigate the transformation of velocities and accelerations among coordinate systems. These latter quantities are used in subse- quent chapters to study the velocity kinematics in Chapter Five, inclu- ding the derivation of the manipulator Jacobian, and also to derive the dynamic equations of motion of rigid manipulators in Chapter Six.
I Since we make extensive lISC of clementary matrix theory, the reader may wish to rc- view Appendix A before beginning this chapter. We wi! Since the inner product is commutative, i. The column vectors of R 6 are of unit length and mutually ortho- gonal Problem It is customary to refer to the set of all 3 x 3 rotation ma- trices by the symbol SO 3.
NOte that hy convention the positive sense for thl. In this case we find it useful to use the more descriptive no- tation Rz,e instead of R 6 to denote the matrix 2. We may also interpret a given rotation matrix as specifying the orientation of the coordinate frame ox lY lZ I relative to the frame oXoYoZo. Projec- ting the unit vectors il,jl, k 1 onto io,jo, k o gives the coordinates of idl' k l in the OXoYoZo frame.
In other words, in- stead of relating the coordinates of a fixed vector with respect to two different coordinate frames, the expression 2.
It represents a coordinate transformation relating the coordi- nates of a point p in two different frames. It gives the orientation of a transformed coordinate frame with respect to a fixed coordinate frame. It is an operator taking a vector P and rotating it to a new vector R P. The particular interpretation of a given rotation matrix R that is being used must then be made clear by the context. It is important for subsequent chapters that the reader understand the material in this section thoroughly before moving on.
Substituting 2. We may interpret Equation 2. Suppose initially that all three of the coordinate frames coincide. Then the ma- trix R is given by 2. The reason is that ro- tation, unlike position, is not a vector quantity and is therefore not subject to the laws of vector addition, and so rotational transformations do not commute in general. Iii Example 2. Many times it is desired to perform a sequence of rotations, each about a given fixed coordinate frame, rather than about successive current frames.
For example we may wish to perform a rotation about the Xo axis followed by a rotation about the Yo and not YI! We will refer to oXoYOZo as the fixed frame. In this case the above composi- tion law is not valid. It turns out that the correct composition law in this case is simply to multiply the successive rotation matrices in the reverse order from that given by 2. Note that the rotations them- selves are not performed in reverse order. Rather they are performed about the fixed frame rather than about the current frame.
Refer to Figure Let Po, PI, and P2 be representations of a vector p. In order to use our previous composition law we need somehow to have the fixed and current frames, in this case Zo and Z I, coincident.
Now, substituting 2,2. We can summarize the rule of composition of rotational transforma- tions by the following recipe. The frame OX2Y2Z2 that results in 2. We are often interested in a rotation about an arbitrary axis in space. We wish to derive the rotation matrix R k,e representing a rotation of e degrees about this axis. There are several ways in which the matrix Rk,e can be derived. Perhaps the simplest way is to rotate the vector k into one of the coor- dinate axes, say Zo, then rotate about Zo by e and finally rotate k back to its original position.
Referring to Figure we see that we can ro- tate k into Zo by first rotating about Zo by -a, then rotating about Yo by - p. Since all rotations are performed relative to the fixed frame oXoYoZo the matrix Rk,e is obtained as 2. Indeed a rigid body possesses at most three rotational degrees-of. In this section we derive three ways in which an arbitrary rotation can he represented using only three independent qu.
The second is rhe Euler Angle representation and the thjnl is the roll- pitch-yaw representati n. Equation 2. However, since the equivalent axis k is given as a unit vector only two of its components are independent.
The third is constrained by the condition that k is of unit length. Therefore, only three independent quantities are required in this representation of a rotation R. Next rotate about the current Y axis by the angle e.
In terms of the basic rotation matrices the resulting rotational transformation Rb can be gener- ated as the product 2. Since the suc- cessive rotations are relative to the fixed frame, the resulting transfor- mation matrix is given by 2.
The end result is the same matrix 2. Thus io,jo,k o are parallel to idl,k l, respectively. Then any point P has representation Po and PI as before. Since the respective coordinate axes in the two frames are parallel the vectors Po and PI are related by 2.
In our case we will never have need for the most general rigid motion, so we assume always that R E SO 3. If we have the two rigid motions 2. In order to represent the transformation 2. Sj' I r dJ. The rationale behind the choice of letters n, s :md a is explained in Chapter Three. Cad 0 0 0 The homogeneous representation 2.
Such transformations in- volve computing derivatives of rotation matrices. By introducing the notion of a skew symmetric matrix it is possible to simplify many of the computations involved. Equation As we will see, 2. The left hand side of Equation 2. The equation says therefore that the matrix representation of 5 a in a coordinate frame rotated by R is the same as the skew symmetric matrix 5 R a corresponding to the vector a rotated by R.
Suppose now that a rotation matrix R is a function of the single vari- able e. Multiplying both sides of 2. It says that computing the deriva- tive of the rotation matrix R is equivalent to a matrix multiplication by a skew symmetric matrix S. The most commonly encountered si- tuation is the case where R is a basic rotation matrix or a product of basic rotation matrices.
Using thiS fact together with Problem it fullows that ;2. In this section WI. An argument identir. Suppose that the homogeneous transformation relating the two frames is time-dependent, so that 2. We may also derive the expression for the relative acceleration in the two coordinate frames as follows. It is always directed toward the axis of rotation and is perpendicular to that axis. In this case it is left as an exercise to show that Equation 2.
The term 2roxRpi is known as the Coriolis ac- celeration. Taking derivatives of both sides of 2. Using the expression 2. The above expression can be extended to any number of coordinate systems. Sketch the initial and final frames. Give a physical interpretation of k.
Find Rkfl. Sketch the initial and fina. What is the direction of the XI axis rdative to the hase frame? Sketch the frame.
A robot is St. A framt: OjX'Y1Zj is fixed to the edge of the table: as shown. A cube measuring 20 cm on a side is placed in tbt: center of the table With frame 02X1Y2Z1 establisht:d at tht: centt:r of the cube ,IS shown. A camera is situat ,;J Jin:ctly ilbove: the center of rhe block 2 m above the tJ. Use this and Problem to verify Equation 2. Alternatively use the fact that R ,. The components of i,j,k, respectively, arc called thc nutation, spin, and precession. XIY1Zl, and O!
X1J'2Z2 are givcn below. The forward kinematics problem can be stated as follows: Given the joint variables of the robot, determine the position and orientation of the end-effector. The joint variables are the angles between the links in the case of revolute or rotational joints, and the link extension in the case of prismatic or sliding joints. The forward kinematics problem is to be contrasted with the inverse kinematics problem, which is studied in the next chapter, and which can be stated as follows: Given a desired position and orientation for the end-effector of the robot, determine a set of joint variables that achieve the desired position and orientation.
The joints can either be very simple, such as a revolute joint or a prismatic joint, or else they can be more complex, such as a ball and socket joint. Recall that a re- volute joint is like a hinge and allows a relative rotation about a single axis, and a prismatic joint permits a linear motion along a single axis, namely an extension or retraction.
In COntrast, a h and socket joint has two dcgn:cs-of-frcedom. Note that the assumption does not involve any real loss of generality, since joints such as a ball and socket joint twO def. With the assumption that each joint has a single dt! The nhjcctive of forward kmematic analysis is to detennine the cumulative effect of tbe entiJ"e set of juint variables.
To do this in a systematic m:mner, one should really intro- duce some conventions. Moreover, they give rise to a universal language with which robot en- gineer! The ;aims are num- bered from 1 to n, and The j -th joint is the point in sp:at. In the case of a revolute jOlOt, q, is the angle of rotOltion, and in lhe t.
Next, a coonliJulte frame is attached rigidly to each link. To he specific, we attach an int:rtial frame to the hase and call it frame O. Then we choose frames 1 through n such that frame i is rigidly 3uached to link i.
This means that, whatever motion the roOOt executes, the coordinates of each point on link i arc constant when expre! The dynamics of a particular experimental setup as abstraction of the rotational degrees of … Expand. View 1 excerpt, cites methods. This chapter presents an image-based Proportional Integral Derivative PID controller for a redundant overactuated planar parallel robot; the control objective is to drive the robot end effector to … Expand.
View 2 excerpts. Robust path following for robot manipulators. Engineering, Computer Science. Path following controllers make the output of a control system approach and traverse a pre-specified path with no a priori time-parametrization. This paper implements a path following controller, … Expand. Highly Influenced. View 7 excerpts, cites background and methods. Dynamics and control of quadrotor with robotic manipulator. We show that the Lagrange dynamics of quadrotor-manipulator systems can be completely decoupled into: 1 the center-of-mass dynamics in E 3 , which, similar to the standard quadrotor dynamics, is … Expand.
View 5 excerpts, cites background. A benchmark study on the planning and control of industrial robots with elastic joints. In this paper, we perform a comparative study of methods for improving motion performance of industrial robots.
The compared methods address trajectory planning, trajectory preshaping, and … Expand. View 1 excerpt, cites background. Robust kinematic control of manipulator robots using dual quaternion representation. Mathematics, Computer Science. Dynamics of Serial Robotic Manipulators. The main objectives of this chapter are a to devise an algorithm for the real-time computed-torque control and b to derive the system of second-order ordinary differential equations ODE … Expand.
In this note, we investigate the adaptive control problem for robot manipulators with both the uncertain kinematics and dynamics.
We propose two adaptive control schemes to realize the objective of … Expand. Robot analysis and control. View 1 excerpt, references background. Robot motion planning. The Kluwer international series in engineering and computer science. View 2 excerpts, references background.
0コメント