10 -18 Robotics and Automation Handbook Using ˆ a and c,wecandefine a local coordinate system on the tool defined by the set of mutually orthogonal unit vectors ˆ e 1 , ˆ e 2 , and ˆ e 3 . The local coordinate system is defined as ˆ e 1 = ˆ a if | ˙ ˆ a|=0 then ˆ e 2 = ˙ ˆ a | ˙ ˆ a| and ˆ e 3 = ˆ e 1 × ˆ e 2 (10.29) if | ˙ ˆ a|=0 and ˙ ˆ a · ˙ c = 0 then ˆ e 3 = ˆ a × ˙ c | ˙ c| and ˆ e 2 = ˆ e 3 × ˆ e 1 Consider, as depicted in Figure 10.15, the circle defined by the intersection of a plane normal to the cutting tool’s axis of rotation ˆ a, offset from the origin of the tool coordinate system c, by the perpendicular distance u. A point p lies in that plane at an angle θ measured from the axis ˆ e 2 . The velocity of point is v = ˙ c + u| ˙ ˆ a| ˆ e 2 −r (u)| ˙ ˆ a| ˆ e 1 cos θ (10.30) The unit normal vector to the surface of revolution at point p is ˆ n = −r (u) ˆ e 1 + ˆ e 2 cos θ + ˆ e 3 sin θ √ 1 +r (u) 2 (10.31) where r (u) = dr dz z=u (10.32) n ^ a ^ e 1 ^ e 3 ^ e 2 ^ Θ Workpiece Local Coordinate System Cutting Tool Local Coordinate System Origin a ^ c c p v FIGURE 10.15 Notation for calculating points on the critical curve. Copyright © 2005 by CRC Press LLC Error Budgeting 10 - 19 Combining Equation (10.30) and Equation (10.31) into Equation (10.24) yields an equation of the form A cos + B sin +C = 0 (10.33) where A =| ˙ ˆ a|r (u)r(u) + ˙ c · ˆ e 2 +| ˙ ˆ a|u (10.34) B = ˙ c · ˆ e 3 (10.35) C =−r (u) ˙ c · ˆ e 1 (10.36) Equation (10.33) admits a closed form solution D = B 2 + A 2 −C 2 (10.37) cos θ = C 2 − AC − B 2 + BD A 2 − AC + B 2 − BD (10.38) sin θ = (A −C )(B − D) A 2 − AC + B 2 − BD (10.39) The results of Equation (10.38) and Equation (10.39) can be used to compute the positions of the points p(t,u) on the surface of the working envelope p(t,u) = c +u ˆ e 1 +r (u)( ˆ e 2 cos θ ± ˆ e 3 sin θ) (10.40) The “±” symbol indicates that there are two solutions for p(t,u). If the tool is cutting a trough, then both solutions are valid. If the tool is contouring counterclockwise as viewed in the positive ˆ e 3 direction, then retain only the solution corresponding to the plus sign. If the tool is contouring clockwise as viewed in the positive ˆ e 3 direction, then retain only the solution corresponding to the minus sign. As is evident from Equation (10.39) above, if A is equal to −C then sin θ is zero, cosθ is one, and θ equals zero. For any values of u such that A − C < 0, that cross section of the tool is not in contact with the workpiece surface. Similarly, if the value of D is imaginary, the results of the calculation can be safely disregarded because the cutting tool is again not in contact with the workpiece. This occurs only when the tool is plunging at a steep angle so that every point cut by that section of the tool is immediately swept away by the cross sections following behind it. The closed formsolution described abovecan beusedin error budgeting byperforming three operations: (1) compute the swept envelope of the cutting tool using a kinematic model with no error motions to determine the nominal machined surface; (2) compute the swept envelope of the cutting tool including error motions to determine the perturbed machined surface; (3) evaluate the accuracy of the machining process by comparing the perturbed and nominal machined surfaces. The procedure described above has proved useful for modeling form grinding, centerless grinding, and “chasing the pin” cylindrical grinding. It should also be useful for ball end milling and flank milling with conical milling cutters. Due to the assumption that the cutter is a perfect surface or revolution, it is most useful for evaluating tolerances of form, profile, location, and size and is probably less useful for surface finish. 10.8 Summary An error budget is a tool for predicting and managing variability in an engineering system. This chapter has reviewed basic theory of probability, tolerances, and kinematics and described a framework for error budgeting based upon those theoretical foundations. The framework presented here is particularly suited to manufacturing systems including robots, machine tools, and coordinate measuring machines. Copyright © 2005 by CRC Press LLC 10 -20 Robotics and Automation Handbook Anerrorbudget must be developedwith greatcare because smallmistakesin theunderlying assumptions or the mathematical implementation can lead to erroneous results. For this reason, error budgets should be kept as simple as possible, consistent with the needs of the task at hand. When error budgets are scoped appropriately,developed rigorously, and consistent with theoretical foundations (e.g., engineering science, mathematics, and probability), they are an indispensable tool for system design. References [1] Donaldson, R.R. (1980). Error Budgets. Technology of Machine Tools, Vol. 5, Machine Tool Task Force, Robert J. Hocken, Chairman, Lawrence Livermore National Laboratory. [2] Slocum, A.H. (1992). Precision Machine Design. Prentice Hall, Englewood Cliffs, NJ. [3] Soons, J.A., Theuws, F.C., and Schellekens, P.H. (1992). Modeling the errors of multi-axis machines: a general methodology. Precision Eng., vol. 14, no. 1, pp. 5–19. [4] Chao,L.M. andYang, J.C.S.(1987). Implementationof ascheme to improve thepositioning accuracy of an articulate robot by using laser distance-measuring interferometry, Precision Eng., vol. 9, no. 4, pp. 210–217. [5] Frey, D.D.,Otto, K.N.,and Pflager,W. (1997). Swept envelopes of cutting tools in integrated machine and workpiece error budgeting. Ann. CIRP, vol. 46, no. 1, pp. 475–480. [6] Frey, D.D., Otto, K.N., and Taketani, S. (2001). Manufacturing system block diagrams and optimal adjustment procedures. ASME J. Manuf. Sci. Eng., vol. 123, no. 1, pp. 119–127. [7] Frey, D.D. and Hykes, T. (1997). A Method for Virtual Machining. U.S. Patent #5,691,909. [8] Treib, T. (1987). Error budgeting — applied to the calculation and optimization of the volumetric error field of multiaxis systems. Ann. CIRP, vol. 36, no. 1, pp. 365–368. [9] Portman, T. (1980). Error summation in the analytical calculation of lathe accuracy. Machines and Tooling, vol. 51, no. 1, pp. 7–10. [10] Narawa, L., Kowalski, M., and Sladek, J. (1989). The influence of kinematic errors on the profile shapes by means of CMM. Ann. CIRP, vol. 38, no. 1, pp. 511–516. [11] Whitney, D.E., Gilbert, O.L., and Jastrzebski, M. (1994). Representation of geometric variations using matrix transforms for statistical tolerance analysis in assemblies. Res. Eng. Design, vol. 6, pp. 191–210. [12] Donmez, A. (1995). A General Methodology for Machine Tool Accuracy Enhancement: Theory, Appli- cation, and Implementation, Ph.D. thesis, Purdue University. [13] Ceglarek, D. and Shi, J. (1996). Fixture failure diagnosis for the autobody assembly using pattern recognition. ASME J. Eng. Ind., vol. 118, no. 1, pp. 55–66. [14] Kurfess, T.R., Banks, D.L., and Wolfson, J.J. (1996). A multivariate statistical approach to metrology. ASME J. Manuf. Sci. Eng., vol. 118, no. 1, pp. 652–657. [15] Drake, A.W. (1967). Fundamentals of Applied Probability Theory. McGraw-Hill, New York. [16] ASME (1983). ANSI Y14.5M — Dimensioning and Tolerancing. American Society of Mechanical Engineering, New York. [17] Kane, V.E. (1986). Process capability indices. J. Qual. Technol., vol. 18, no. 1, pp. 41–52. [18] Harry, M.J. and Lawson, J.R. (1992). Six Sigma Producibility Analysis and Process Characterization. Addison-Wesley, Reading, MA. [19] Phadke, M.S. (1989). Quality Engineering Using Robust Design. Prentice Hall, Englewood Cliffs, NJ. [20] Denavit, J. and Hartenberg, R. (1955). A kinematic notation for lower pair mechanisms based on matrices. J. Appl. Mech, vol. 1, pp. 215–221. [21] Bryan, J.B. (1989). The Abb ´ e principle revisited — an updated interpretation. Precision Eng., vol. 1, no. 3, pp. 129–132. [22] Lin, P.D. and Ehmann, K.F. (1993). Direct volumetric error evaluation for multi-axis machines. Int. J. Machine Tools Manuf., vol. 33, no. 5, pp. 675–693. [23] CIRP (1978). A proposal for defining and specifying the dimensional uncertainty of multiaxis measuring machines. Ann. CIRP, vol. 27, no. 2, pp. 623–630. Copyright © 2005 by CRC Press LLC Error Budgeting 10 - 21 [24] Shen, Y.L. and Duffie, N.A. (1993). Comparison of combinatorial rules for machine error budgets. Ann. CIRP, vol. 42, no. 1, pp. 619–622. [25] Hocken, R.J. and Machine Tool Task Force (1980). Technology of Machine Tools, UCRL-52960-5, Lawrence Livermore National Laboratory, University of California, Livermore, CA. [26] Wang, W.P. and Wang, K.K. (1986). Geometric modeling for swept volume of moving solids. IEEE Comput. Graphics Appl., vol. 6, no. 12, pp. 8–17. Copyright © 2005 by CRC Press LLC 11 Design of Robotic End Effectors Hodge Jenkins Mercer University 11.1 Introduction 11.2 Process and Environment System Design • Special Environments 11.3 Robot Attachment and Payload Capacity Integrated End Effector Attachment • Attachment Precision • Special End Effector Locations • Wrist Compliance: Remote Compliance Centers • Payloads • Payload Force Analysis 11.4 Power Sources Compressed Air • Vacuum • Hydraulic Fluid Power • Electrical Power • Other Actuators 11.5 Gripper Kinematics Parallel Axis/Linear Motion Jaws • Pivoting/Rotary Action Jaws • Four-Bar Linkage Jaws • Multiple Jaw/Chuck Style • Articulating Fingers • Multi-Component End Effectors 11.6 Grasping Modes, Forces, and Stability Grasping Stability • Friction and Grasping Forces 11.7 Design Guidelines for Grippers and Jaws Gripper and Jaw Design Geometry • Gripper Design Procedure • Gripper Design: Case Study • Gripper Jaw Design Algorithms • Interchangeable End Effectors • Special Purpose End Effectors/Complementary Tools 11.8 Sensors and Control Considerations Proximity Sensors • Collision Sensors • Tactile Feedback/Force Sensing • Acceleration Control for Payload Limits • Tactile Force Control 11.9 Conclusion 11.1 Introduction Aside from the robot itself, the most critical device in a robotic automation system is the end effector. Basic grasping end effector forms are referred to as grippers. Designs for end effectors are as numerous as the applications employing robots. End effectors can be part of the robot’s integral design or added-on to the base robot. The design depends on the particular robot being implemented, objects to be grasped, tasks to be performed, and the robot work environment. This chapter outlines many of the design and selection decisions of robotic end effectors. First, process and environment considerations are discussed. Robot considerations including power, joint compliance, payload capacity, and attachment are presented. Sections reviewing basic end effector styles and design Copyright © 2005 by CRC Press LLC 12 Sensors and Actuators Jeanne Sullivan Falcon National Instruments 12.1 Encoders Rotary and Linear Incremental Encoders • Tachometer • Quadrature Encoders • Absolute Encoders 12.2 Analog Sensors Analog Displacement Sensors • Strain • Force and Torque • Acceleration 12.3 Digital Sensors Switches as Digital Sensors • Noncontact Digital Sensors • Solid State Output • Common Uses for Digital Sensors 12.4 Vision 12.5 Actuators Electromagnetic Actuators • Fluid Power Actuators 12.1 Encoders 12.1.1 Rotary and Linear Incremental Encoders Incremental encoders are the most common feedback devices for robotic systems. They typically output digital pulses at TTL levels. Rotary encoders are used to measure the angular position and direction of a motor or mechanical drive shaft. Linear encoders measure linear position and direction. They are often used in linear stages or in linear motors. In addition to position and direction of motion, velocity can also be derived from either rotary or linear encoder signals. In a rotary incremental encoder, a glass or metal disk is attached to a motor or mechanical drive shaft. The disk has a pattern of opaque and transparent sectors known as a code track. A light source is placed on one side of the disk and a photodetector is placed on the other side. As the disk rotates with the motor shaft, the code track interrupts the light emitted onto the photodetector, generating a digital signal output (Figure 12.1). The number of opaque/transparent sector pairs, also known as line pairs, on the code track corresponds to the number of cycles the encoder will output per revolution. The number of cycles per revolution (CPR) defines the base resolution of the encoder. 12.1.2 Tachometer An incremental encoder with a single photodetector is known as a tachometer encoder. The frequency of the output pulses is used to indicate the rotational speed of the shaft. However, the output of the single-channel encoder cannot give any indication of direction. Copyright © 2005 by CRC Press LLC Sensors and Actuators 12 - 5 is determined by the position of the core within the tube. The magnitude of the output of the signal is a function of the distance of the core from the primary coil, and the phase of the output signal is a function of the direction of the core from the primary coil — towards one secondary coil or the other as shown in the figure. LVDTsensorscan be usedinapplicationswith large travelrequirements.However,mechanicalalignment along the direction of travel is important for this type of sensor. 12.2.1.3 Resolvers A resolver is essentially a rotary transformer that can provide absolute position information over one revo- lution. The resolver consists of a primary winding located on the rotor shaft and a two secondary windings located on the stator. The secondary windings are oriented 90 ◦ relative to each other. Energy is applied to the primary winding on the rotor. As the rotor moves, the output energy of the secondary windings varies sinusoidally. Resolvers are an alternative to encoders for joint feedback in robotic applications. 12.2.1.4 Inductive (Eddy Current) Inductive sensors are noncontact sensors and can sense the displacement of metallic (both ferrous and nonferrous) targets. Themost common type of inductive sensor used in robotics isthe eddy current sensor. The sensor typically consists of two coils of wire: an active coil and a balance coil, with both driven with a high frequencyalternating current.When ametallic targetis placed near the active sensor coil, the magnetic field from the active coil induces eddy currents in the target material. The eddy currents are closed loops of current and thus create their own magnetic field. This field causes the impedance of the active coil to change. The active coil and balance coil are both included in a bridge circuit. The impedance change of the active coil can be detected by measuring the imbalance in the bridge circuit. Thus, the output of the sensor is dependent upon the displacement of the target relative to the face of the sensor coil. The effective depth of the eddy currents in the target material, δ, is given by δ = 1 π f µσ where f is the excitation frequency of the coil, µ is the magnetic permeability of the target material, and σ is the conductivity of the target material. In order to ensure adequate measurement, the target material must be three times as thick as the effective depth of the eddy currents. In general, the linear measurement range for inductive sensors is approximately 30% of the sensor active coil diameter. The target area must be at least as large as the surface area of the sensor probe. It is possible to use curved targets, but their diameter should be three to four times the diameter of the sensor probe. In addition, the sensor signal is weaker for ferrous target materials compared with nonferrous target materials. This can lead to a reduced measurement range and so this should be investigated with the sensor manufacturer. Inductive sensors can sense through nonmetallic objects or nonmetallic contamination. However, if measurement of a nonmetallic target displacement is required, then a segment of metallic material must be attached to the target. 12.2.1.5 Capacitive Capacitive displacement sensors are another type of noncontact sensor and are capable of directly sensing both metallic and nonmetallic targets. These sensors are designed using parallel plate capacitors. The capacitance is given by C = ε r ε 0 A d whereε r is therelative permittivity (dielectric constant) ofthe materialbetween theplates, ε 0 is theabsolute permittivity of free space, A is the overlapping area between the plates, and d is the distance between the plates. Copyright © 2005 by CRC Press LLC 12 -6 Robotics and Automation Handbook d A sensor plate sensor plate targettarget C C FIGURE 12.6 Distance and area variation in capacitive sensor measurement. In displacement sensor designs, a capacitive sensor typically incorporates one of the capacitor plates within its housing and the target forms the other plate of the capacitor. The sensor then operates on the principle that the measured capacitance is affected by variation in the distance d or the overlapping area A between the plates. The capacitance is measured by detecting changes in an oscillatory circuit that includes the capacitor. Figure 12.6 shows the distance and area variation methods for capacitive displacement measurement. Fordetection ofnonmetallic targets,a stationary metallic reference isused as the externalcapacitor plate. The presence of the nonmetallic target in the gap between the sensor and the stationary metallic reference will change the permittivity and thus affect the measured capacitance. The capacitance will be determined by the thickness and location of the nonmetallic target. Figure 12.7 shows the dielectric variation approach for capacitive displacement measurement. In general, the linear measurement range for capacitive sensors is approximately 25% of the sensor diameter. The target should be 30%larger than thesensor diameter for optimum performance.In addition, environmental contamination can change the dielectric constant between the sensor and the target, thus reducing measurement accuracy. 12.2.1.6 Optical Sensors Optical sensors provide another means of noncontact displacement measurement. There are several types which are commonly used in robotics: optical triangulation, optical time-of-flight, and photoelectric. 12.2.1.6.1 Optical Triangulation Optical triangulation sensors use a light emitter, either a laser or an LED, in combination with a light receiver to sense the position of objects. Both the emitter and receiver are contained in the same housing as shown in Figure 12.8. The emitter directs light waves toward a target. These are reflected off the target, through a lens, to the receiver. The location of the incident light on the receiver is used to determine the position of the target in relation to the sensor face. The type of receiver used may be a position sensitive detector (PSD) or a pixelized array device such as a charge coupled device (CCD). The PSD receiver generates a single analog output and has a faster response time than the output pixelized array device because less post-processing is required. It is also typically smaller so that the overall sensor size will be smaller. Pixelized array devices, however, are useful when the surface of the target is irregular or transparent. non-metallic target metallic reference sensor plate C FIGURE 12.7 Dielectric variation in capacitive sensor measurement. Copyright © 2005 by CRC Press LLC . 10 -1 8 Robotics and Automation Handbook Using ˆ a and c,wecandefine a local coordinate system on the tool defined by the set of mutually orthogonal unit vectors ˆ e 1 , ˆ e 2 , and ˆ e 3 Sci. Eng., vol. 11 8, no. 1, pp. 652–657. [15 ] Drake, A.W. (19 67). Fundamentals of Applied Probability Theory. McGraw-Hill, New York. [16 ] ASME (19 83 ). ANSI Y14.5M — Dimensioning and Tolerancing New York. [17 ] Kane, V.E. (19 86 ). Process capability indices. J. Qual. Technol., vol. 18 , no. 1, pp. 41 52. [ 18 ] Harry, M.J. and Lawson, J.R. (19 92). Six Sigma Producibility Analysis and Process