The Conceptual Design of Robotic Architectures using Complexity Rules

Transcription

1 The Conceptual Design of Robotic Architectures using Complexity Rules Waseem A. Khan and J. Angeles Centre for Intelligent Machines & Department of Mechanical Engineering McGill University Montreal, Quebec, Canada W.A. Khan and J. Angeles (CIM) 1 / 27

2 Outline 1 Introduction 2 Performance Evaluation of LKPs 3 Performance Evaluation of Kinematic Chains 4 A Complexity-based Framework 5 Example 6 Conclusions W.A. Khan and J. Angeles (CIM) 2 / 27

3 Introduction Problem Statement Problem Statement Scope To develop a concept-evaluation framework for robot design Requirements Should be quantitative Should lead to simplicity Should provide a small, effective set of fine-tuning parameters Should lend itself to extensions Benefits Could provide tools to complement the robot designer s judgement; and support the contemporary efforts towards the automatic synthesis of robot structures and mechatronic systems W.A. Khan and J. Angeles (CIM) 3 / 27

4 Introduction Problem Statement Problem Statement Scope To develop a concept-evaluation framework for robot design Requirements Should be quantitative Should lead to simplicity Should provide a small, effective set of fine-tuning parameters Should lend itself to extensions Benefits Could provide tools to complement the robot designer s judgement; and support the contemporary efforts towards the automatic synthesis of robot structures and mechatronic systems W.A. Khan and J. Angeles (CIM) 3 / 27

5 Introduction Problem Statement Problem Statement Scope To develop a concept-evaluation framework for robot design Requirements Should be quantitative Should lead to simplicity Should provide a small, effective set of fine-tuning parameters Should lend itself to extensions Benefits Could provide tools to complement the robot designer s judgement; and support the contemporary efforts towards the automatic synthesis of robot structures and mechatronic systems W.A. Khan and J. Angeles (CIM) 3 / 27

6 Introduction Problem Statement Problem Statement (Cont d) Challenge To quantify the quality of the design alternatives in the absence of a mathematical model. Information available: type and number of joints, relative orientation of neighbouring joints, number of loops (parallel chains), type and diversity of actuators, etc. A robot concept W.A. Khan and J. Angeles (CIM) 4 / 27

7 Introduction Problem Statement Problem Statement (Cont d) Challenge To quantify the quality of the design alternatives in the absence of a mathematical model. Information available: type and number of joints, relative orientation of neighbouring joints, number of loops (parallel chains), type and diversity of actuators, etc. A robot concept W.A. Khan and J. Angeles (CIM) 4 / 27

8 Introduction Research Issues Definitions Kinematic Pair A kinematic pair is the coupling of two rigid bodies, called links Lower Kinematic Pair A lower kinematic pair (LKP) is obtained when the two links share a common surface. LKPs are among the basic ingredients of a robot structure Higher Kinematic Pair A higher kinematic pair (HKP) is obtained when the coupling takes place along a common line or a common point Kinematic Chain A kinematic chain is the result of the coupling of links via kinematic pairs W.A. Khan and J. Angeles (CIM) 5 / 27

9 Introduction Research Issues Definitions Kinematic Pair A kinematic pair is the coupling of two rigid bodies, called links Lower Kinematic Pair A lower kinematic pair (LKP) is obtained when the two links share a common surface. LKPs are among the basic ingredients of a robot structure Higher Kinematic Pair A higher kinematic pair (HKP) is obtained when the coupling takes place along a common line or a common point Kinematic Chain A kinematic chain is the result of the coupling of links via kinematic pairs W.A. Khan and J. Angeles (CIM) 5 / 27

10 Introduction Research Issues Definitions Kinematic Pair A kinematic pair is the coupling of two rigid bodies, called links Lower Kinematic Pair A lower kinematic pair (LKP) is obtained when the two links share a common surface. LKPs are among the basic ingredients of a robot structure Higher Kinematic Pair A higher kinematic pair (HKP) is obtained when the coupling takes place along a common line or a common point Kinematic Chain A kinematic chain is the result of the coupling of links via kinematic pairs W.A. Khan and J. Angeles (CIM) 5 / 27

11 Introduction Research Issues Definitions Kinematic Pair A kinematic pair is the coupling of two rigid bodies, called links Lower Kinematic Pair A lower kinematic pair (LKP) is obtained when the two links share a common surface. LKPs are among the basic ingredients of a robot structure Higher Kinematic Pair A higher kinematic pair (HKP) is obtained when the coupling takes place along a common line or a common point Kinematic Chain A kinematic chain is the result of the coupling of links via kinematic pairs W.A. Khan and J. Angeles (CIM) 5 / 27

12 Introduction Research Issues Definitions (Cont d) Examples The six lower kinematic pairs An open and a closed kinematic chain W.A. Khan and J. Angeles (CIM) 6 / 27

13 Introduction Research Issues Problem Statement (Cont d) Research Issues Performance evaluation of different kinematic pairs Performance evaluation of kinematic chains Decision-making framework Our Approach The objective, to compare robot topologies at the conceptual stage, is essentially based on simplicity, or its complement, complexity Here we liken design complexity to the diversity content of a design variant W.A. Khan and J. Angeles (CIM) 7 / 27

14 Introduction Research Issues Problem Statement (Cont d) Research Issues Performance evaluation of different kinematic pairs Performance evaluation of kinematic chains Decision-making framework Our Approach The objective, to compare robot topologies at the conceptual stage, is essentially based on simplicity, or its complement, complexity Here we liken design complexity to the diversity content of a design variant W.A. Khan and J. Angeles (CIM) 7 / 27

15 Performance Evaluation of LKPs Loss of Regularity Performance Evaluation of LKPs Which criterion can be used? Some characteristic of the common surface of a given LKP may be plausible Definition Loss of Regularity κ rms κ rms LOR κ rms 2 κ rms 2 r.m.s. of the two principal curvatures at a point of the surface the derivative of κ rms with respect to a geometric dimensionless parameter σ, e.g., arc length The LOR is inspired from Taguchi s loss function The LOR measures the diversity of the curvature distribution of the given surface LOR for surfaces with G 1 or G 2 discontinuities W.A. Khan and J. Angeles (CIM) 8 / 27

16 Performance Evaluation of LKPs Loss of Regularity Performance Evaluation of LKPs Which criterion can be used? Some characteristic of the common surface of a given LKP may be plausible Definition Loss of Regularity κ rms κ rms LOR κ rms 2 κ rms 2 r.m.s. of the two principal curvatures at a point of the surface the derivative of κ rms with respect to a geometric dimensionless parameter σ, e.g., arc length The LOR is inspired from Taguchi s loss function The LOR measures the diversity of the curvature distribution of the given surface LOR for surfaces with G 1 or G 2 discontinuities W.A. Khan and J. Angeles (CIM) 8 / 27

17 Performance Evaluation of LKPs Loss of Regularity Performance Evaluation of LKPs Which criterion can be used? Some characteristic of the common surface of a given LKP may be plausible Definition Loss of Regularity κ rms κ rms LOR κ rms 2 κ rms 2 r.m.s. of the two principal curvatures at a point of the surface the derivative of κ rms with respect to a geometric dimensionless parameter σ, e.g., arc length The LOR is inspired from Taguchi s loss function The LOR measures the diversity of the curvature distribution of the given surface LOR for surfaces with G 1 or G 2 discontinuities W.A. Khan and J. Angeles (CIM) 8 / 27

18 Performance Evaluation of LKPs Loss of Regularity Notes The complexity of LKPs can be based on the regularity of the common surface Common surface of each LKP is assigned as of minimum LOR Common surface should be G 2 continuous There exist infinite choices for the foregoing surfaces Surfaces generated from algebraic curves of minimum degree are expected to exhibit the minimum LOR W.A. Khan and J. Angeles (CIM) 9 / 27

19 Performance Evaluation of LKPs Loss of Regularity Notes The complexity of LKPs can be based on the regularity of the common surface Common surface of each LKP is assigned as of minimum LOR Common surface should be G 2 continuous There exist infinite choices for the foregoing surfaces Surfaces generated from algebraic curves of minimum degree are expected to exhibit the minimum LOR W.A. Khan and J. Angeles (CIM) 9 / 27

20 Performance Evaluation of LKPs Loss of Regularity Notes The complexity of LKPs can be based on the regularity of the common surface Common surface of each LKP is assigned as of minimum LOR Common surface should be G 2 continuous There exist infinite choices for the foregoing surfaces Surfaces generated from algebraic curves of minimum degree are expected to exhibit the minimum LOR W.A. Khan and J. Angeles (CIM) 9 / 27

21 Performance Evaluation of LKPs Loss of Regularity Notes The complexity of LKPs can be based on the regularity of the common surface Common surface of each LKP is assigned as of minimum LOR Common surface should be G 2 continuous There exist infinite choices for the foregoing surfaces Surfaces generated from algebraic curves of minimum degree are expected to exhibit the minimum LOR W.A. Khan and J. Angeles (CIM) 9 / 27

22 Performance Evaluation of LKPs Loss of Regularity Notes The complexity of LKPs can be based on the regularity of the common surface Common surface of each LKP is assigned as of minimum LOR Common surface should be G 2 continuous There exist infinite choices for the foregoing surfaces Surfaces generated from algebraic curves of minimum degree are expected to exhibit the minimum LOR W.A. Khan and J. Angeles (CIM) 9 / 27

23 Performance Evaluation of LKPs LOR of LKPs LOR of the Revolute Pair A polynomial is a plausible candidate (the simplest one!) Can be blended smoothly with a cylinder at its ends (a) (b) G: the generatrix of the surface of revolution (c) LORR = 10.3 W.A. Khan and J. Angeles (CIM) 10 / 27

24 Performance Evaluation of LKPs LOR of LKPs LOR of remaining five LKPs Prismatic Pair Planar Pair Helical Pair The LOR of cylindrical and spherical joints is zero W.A. Khan and J. Angeles (CIM) 11 / 27

25 Performance Evaluation of LKPs Geometric Complexity of LKPs Geometric Complexity of LKPs Definition The geometric complexity of the lower kinematic pairs is defined as: K G x LOR x LOR max, LOR max = LOR P = 19.7 Description Loss of regularity Geometric complexity male female mean K G R C P H F S Table: Geometric complexity of the six lower kinematic pairs W.A. Khan and J. Angeles (CIM) 12 / 27

26 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

27 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

28 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

29 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

30 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

31 Performance Evaluation of Kinematic Chains Performance Attributes Some Performance Attributes The kinetostatic performance depends on the robot Jacobian J, which in turn depends on link dimensions and robot posture. Complete information is not available at the conceptual stage! The elastostatic performance can be improved by increasing the stiffness of the robot structure The elastodynamic performance can be improved by increasing the stiffness, by decreasing the mass of the robot, or even by a combination of both Agility refers to the property of a robot to achieve high and accurate operational speeds. The elastodynamic performance is an indicator of the agility of the robot Workspace of the robot Life-cycle cost W.A. Khan and J. Angeles (CIM) 13 / 27

32 Performance Evaluation of Kinematic Chains Knowledge-based Rules A Set of Rules For high elastostatic and agility performances, it is necessary to select topologies that have higher probability of being stiff and lightweight We can use knowledge-based rules to relate the information available at the conceptual design stage to the performance attributes The Relation Table of Knowledge-based Rules Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 The above rules are applicable to serial as well as parallel robots W.A. Khan and J. Angeles (CIM) 14 / 27

33 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R1.1 The number of joints in a robot should be minimized to increase stiffness W.A. Khan and J. Angeles (CIM) 15 / 27

34 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R2.2 Increasing the number of loops increases the manufacturing cost W.A. Khan and J. Angeles (CIM) 15 / 27

35 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R The six lower kinematic pairs are, in order of decreasing preference: cylindrical, spherical, revolute, screw, planar and prismatic W.A. Khan and J. Angeles (CIM) 15 / 27

36 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R2.4 Increasing the diversity in geometric constraints between joints increases the manufacturing cost W.A. Khan and J. Angeles (CIM) 15 / 27

37 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R3.2 Increasing the diversity in geometric constraints between joints increases the manufacturing cost W.A. Khan and J. Angeles (CIM) 15 / 27

38 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R4.1 Increasing the number of joints decreases agility W.A. Khan and J. Angeles (CIM) 15 / 27

39 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R4.2 Addition of loops to allow actuator(s) placed closer to the base increases agility W.A. Khan and J. Angeles (CIM) 15 / 27

40 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 R4.5 In robotics, the use of hydraulic actuators increases agility W.A. Khan and J. Angeles (CIM) 15 / 27

41 Performance Evaluation of Kinematic Chains Knowledge-based Rules Some Rules The Relation Table of Knowledge-based Rules (Cont d) Number Number Type Joint Type Diversity of of of config- of of joints loops joints uration actuators actuators Stiffness R1.1 R R1.5 - Life-Cycle R2.1 R2.2 R2.3.1 R2.4 R2.5 R2.6 Cost R2.3.2 Workspace - R3.2 R Volume R3.3.2 Agility R4.1 R R4.5 Notice that some rules are antagonistic among themselves W.A. Khan and J. Angeles (CIM) 15 / 27

42 A Complexity-based Framework Complexity-based Rules A robot architecture is minimally complex if it abides by the previous rules We define rules that rely on the previously mentioned knowledge-based rules Six aspects of robot complexity are defined in this vein: Joint-number complexity Loop complexity Joint-type complexity Link diversity Actuator-type complexity Actuator diversity W.A. Khan and J. Angeles (CIM) 16 / 27

43 A Complexity-based Framework Joint-number and -type Complexity Joint-number Complexity K N N q N K N = 1 exp( q N N); K N [0, 1] the number of joints used in the topology at hand the resolution parameter Joint-type Complexity K J A convex combination of the form K J = 1 n (n RK G R + n P K G P + n C K G C + n F K G F + n S K G S + n H K G H ) n x K G x n the number of pair type x the geometric complexity of pair type x total number of pairs W.A. Khan and J. Angeles (CIM) 17 / 27

44 A Complexity-based Framework Loop and Actuator-type Complexities Loop Complexity K L l l m q L K L = 1 exp( q L L); L = l l m ; K L [0, 1] number of kinematic loops the minimum number of loops required the resolution parameter Application: Concept evaluation of parallel manipulators Actuator-type Complexity K A a a m K A = 1 exp( q A A); A = a a m number of electromagnetic actuators the minimum number of electromagnetic actuators allowed W.A. Khan and J. Angeles (CIM) 18 / 27

45 A Complexity-based Framework Link and Actuator Diversity Link Diversity K B Figure: Binary tree displaying possible link topologies W.A. Khan and J. Angeles (CIM) 19 / 27

46 A Complexity-based Framework Link and Actuator Diversity Link Diversity K B c M i B B max b i K B = B B max ; B = c b i log 2 (b i ); i=1 b i M i c i=1 M i number of distinct joint-constraint types used in a concept number of instances of each type of joint-constraints the entropy of the link topology maximum possible entropy, i.e., B max = log 2 (5) = 2.32 bits frequency of occurrence of a link topology Actuator Diversity K H d N i H max φ i K H = H H max ; H = d φ i log 2 (φ i ); i=1 φ i N i d i=1 N i number of distinct actuators the number of instances of each type or specification maximum possible entropy, i.e., H max = log 2 ( d i=1 N i) frequency of occurrence of an actuator W.A. Khan and J. Angeles (CIM) 20 / 27

47 A Complexity-based Framework Total Complexity of a Kinematic Chain The Total Complexity K of a Kinematic Chain A convex combination of the form K = w N K N + w L K L + w J K J + w B K B + w A K A + w H K H K [0, 1] w N + w L + w J + w B + w A + w H = 1 W.A. Khan and J. Angeles (CIM) 21 / 27

48 Example Example: Comparison of Given Concepts Concept 1: A six-dof 2PT-1PR six-dof hybrid robot Concatenation of three parallel subsystems The pan-tilt and the pan-roll mechanisms substitute a set of two mutually perpendicular revolutes in series Two identical motors drive each PT/PR mechanism W.A. Khan and J. Angeles (CIM) 22 / 27

49 Example Example: Comparison of Given Concepts (Cont d) An embodiment of Concept 1 (2PT-1PR robot) W.A. Khan and J. Angeles (CIM) 23 / 27

50 Example Example: Comparison of Given Concepts (Cont d) Concept 2 The PUMA robot W.A. Khan and J. Angeles (CIM) Concept 3 The DIESTRO robot 24 / 27

51 Example The DH-parameters of the three concepts at hand C1 C2 C3 Joint a i b i α i a i b i α i a i b i α i m m deg m m deg m m deg b 1 90 a a a 2 b 2 0 a a b 3 90 a a a 90 4 a b 4-90 a a a a 90 6 a 6 b 6 α 6 a 6 b 6 α 6 a a α 6 Complexities of the two concepts under study K N K L K J K B K A K H C C C K C1 = 0.46; K C2 = 0.80; K C3 = 0.5 W.A. Khan and J. Angeles (CIM) 25 / 27

52 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

53 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

54 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

55 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

56 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

57 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

58 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

59 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

60 Conclusions Conclusions A framework to evaluate the complexity of candidate robotic architectures at the conceptual stage was proposed Several expert rules were outlined Six complexity-based rules along with their formulation were proposed The total complexity was found by means of a convex combination of the different types of complexity A comparison among three concepts was provided W.A. Khan and J. Angeles (CIM) 26 / 27

61 Conclusions Thank You! W.A. Khan and J. Angeles (CIM) 27 / 27