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A linkage based solution approach for determining 6 axis serial robotic travel path feasibility.


When performing trajectory planning for robotic applications, there are many aspects to consider, such as the reach conditions, joint and end-effector velocities, accelerations and jerk conditions, etc. The reach conditions are dependent on the end-effector orientations and the robot kinematic structure. The reach condition feasibility is the first consideration to be addressed prior to optimizing a solution. The 'functional' work space or work window represents a region of feasible reach conditions, and is a sub-set of the work envelope. It is not intuitive to define. Consequently, 2D solution approaches are proposed. The 3D travel paths are decomposed to a 2D representation via radial projections. Forward kinematic representations are employed to define a 2D boundary curve for each desired end effector orientation. The feasible region for all orientations is determined by the use of Boolean intersections of the boundary curves or by overlapping valid regions in which points are placed. Combining the tool path and functional work space regions allow designers to readily visualize regions of concern. A KUKA and Comau robot are used to illustrate the methodology.

CITATION: Urbanic, R., Hedrick, R., and Djuric, A., "A Linkage Based Solution Approach for Determining 6 Axis Serial Robotic Travel Path Feasibility," SAE Int. J. Mater. Manf. 9(2):2016,


The development of a wide array of automation strategies and industrial manipulators has facilitated manufacturing operations such as assembly, inspection, material handling, grinding, painting, applying sealant, and so forth. Many tasks are repetitive and are performed in hazardous environments (i.e., a paint booth). A comprehensive set of robot structures have since been designed and built to fulfill the various industry requirements. These multi-Degrees of Freedom (DOF) structures are highly complex in their form and control. Most manipulators used in the industry today are articulated with a combination of six rotational joints linked serially. These robotic arms are inspired from human arms and their ability to rotate, position, and orient hands as shown in Figure 1 [1].

There are several distinct 'standard' kinematic structure families: for example, the ABB and the Fanuc robot family types vary based on the twist angle. This 6 axis serial link structural form provides the manipulators with a great deal of flexibility, and dexterity. In several environments, manufacturing cells are utilized, where machines and material handling systems (i.e. conveyors) are integrated with several 6 axis industrial robots, allowing designers to rearrange system elements for new product variants and/or address volume changes. Significant research has been performed with respect to modelling industrial robots, and robotic systems. Much of this research focuses on the development and optimization of a manipulator design and its functionality within an industrial setting. Due to the kinematic structures of the serial six axis industrial robots, there are challenges related to identifying the kinematic singularities, the complex inverse kinematic solution(s), trajectory planning, and travel path validation for a design configuration.

Various simulation tools allow designers to experiment with various solution approaches. Typically, the Denavit-Hartenberg notation (i.e., DH parameters) is presented to provide the background for the subsequent analyses, as the DH parameters are commonly used in the robotics domain. They provide a standard methodology to write the kinematic equations of a manipulator or end-effector. Each joint in a serial kinematic chain is assigned a coordinate frame. Using the DH notation [2], 4 parameters are needed to describe how a frame relates to a previous frame as shown in Figure 2. The kinematic structure for a KUKA KR robot family is shown in Figure 3, and the Comau family is shown in Figure 4.

After assigning coordinate frames the four D-H parameters can be defined as following:

* [a.sub.1] - Link length is the distance along the common normal between the joint axes

* [[omega].sub.i] -Twist angle is the angle between the joint axes

* [[theta].sub.i] -Joint angle is the angle between the links

* [d.sub.i] - Link offset is the displacement, along the joint axes between the links

The numerical values for the general KUKA KR D-H parameters are [d.sub.1]= -865, [d.sub.4]=1200, [d.sub.6]=-210, [d.sub.1]= -410, [d.sub.2]=1000, and [d.sub.3] = 45. The D-H parameters for KUKA KR and Comau family of robots are summarized in Tables 1 and 2. The similarities and differences can be readily highlighted. The nomenclature used for the end-effector orientation angle for this work is presented in Figure 5.

The workspace for different configurations and degrees of freedom has been extensively analyzed [3,4,5,6,7,8,9,10,11,12,13,14] for several diverse applications and differing manipulator types. However, the workspace or work envelope region does not ensure the feasibility of reaching a point in space for an end-effector orientation. The reach conditions are dependent on both the end-effector orientations and the robot kinematic structure, and the functional space needs to be determined.

The D-H parameters can be employed in algorithms to determine the bounded reachable space for a given end-effector length and orientation, but this can also be determined by reducing the complex 3D problem into a series of 2D problems [15, 16]. The forward kinematics are solved by fixing [[theta].sub.4] and [[theta].sub.6]. This results in the [[theta].sub.2], [[theta].sub.3], [[theta].sub.5] rotation angles being analyzed via forward kinematics.

A 2D feasible region can be determined, and then revolved around [[theta].sub.1] (Figure 6). A complex 3D travel path can also be reduced to a 2D data set using radial projections from an origin point. This addresses the influence of [[theta].sub.1] in 2D space. Combining these 2D solutions (Figure 7) the feasibility related to reach for a position-orientation set for a travel path (or travel path set) can be readily evaluated.

The derived solution approach developed using various CAD/CAM software tools (Mastercam[R], Rhinoceros[R] and Grasshopper[R]), Workspace 5 (a robotics simulation tool), MATLAB, and Excel spreadsheet tools.


Trajectory Projections

There are many software tools and optimization strategies are employed to create travel paths, and more are continuing to be developed [17,18,19]. Trajectory planning for coordinated motion of a robot for a particular task, within a cell containing multiple robots or a rail or positioning table, conveyors, and so forth are being developed and improved upon. However, the solution depends on the initial positioning and orientation, as well as criteria with respect to minimizing joint wrap, energy requirements, velocity changes, least travel time, etc. The inverse kinematics are challenging due to redundancies associated with joints 4 and 6 in addition to the' elbow up or down' and other joint 'refection' solutions. However, prior to optimizing a solution, it is essential to determine whether a solution is feasible. To quickly determine this, the travel paths are radially projected (or squashed) onto a plane. The general procedure is:

* Determine reach points or

** Create travel path (curves)

** Determine segment length [DELTA]l

** Segment a curve path into points by [DELTA]l

** Create points at the segment end points

* Establish a reference point vector ([DELTA][x.sub.r], [DELTA][y.sub.r], [DELTA] [z.sub.r]) from the joint 1 (0,0,0) to the part (0,0,0)

* Determine Cartesian x, y, z values of the points with respect to the reference point using the Pythagorean Theorem

* Calculate the 2D 'radial projection' equivalent Cartesian x, y, 0 values of the points with respect to the reference point

* Compare results to the functional work space boundaries Consider the cube shown in Figure 8 (a). The reference position is 0,0,0, and the initial z depth is 8 units. From the reference point (0,0,0), the radial projection of point (0, 10, 8) is (8, 10, 0) (Figure 8 (b)). The resulting corner points for this reference point are shown in Figure 8 (c).

Keeping the same block size, but offsetting the block two units (Figure 9 (a)) provides a different reach condition (Figure 9 (b)). The boundary box for the original point set is much wider due to the origin position, as shown in Figure 9 (c).

For a curve based travel path, positions on the curve need to be determined and then projected. A piece-wise linear model is created. The granularity is situation dependent. An example of a curve complete with points and their projected positions is shown in the Appendix (Figures 24, 25, 26, 27). This can be extended for all curves or curve sets, i.e., multiple operations such as those for a laser cladding travel path, which are presented in the case studies section.

The end effector orientation may vary for a travel path. A component may be picked in one end effector orientation and placed at another, or the orientation may be at a constant angle to a curve, but the curve wraps (Figure 10). Therefore, all the reachable regions need to be determined. Additional constraints can be added to the basic reach feasibility: no joint wrap, elbow up, etc. Additional constrains can be applied once the basic reach conditions are established. Different methodologies are described in the next section.

Empirical Point Generation Methodology

One basic empirical method to determine the reach space for any orientation angle [phi] is to increment the joints from the minimum joint limit to the maximum by an increment parameter [DELTA] after fixing [[theta].sub.1] to 0 [15]. The end point is determined via the forward kinematics, and stored. A general process flow is developed for 2, 3, 5 or 6 axis articulated robots. Once this extensive point set is generated, sort strategies can be implemented to determine the boundary region points (Figure 10), and curve fitting methods employed to determine the bounded regions.

To initiate the sorting and boundary identification process, the points are sorted in y, and then x. The start point is at the minimum x coordinate for the minimum y value. Scanning from left to right, a boundary edge will occur when there is a positive change in y coordinate values for a large negative change in x coordinate values. After a 'left to right' scan is completed, the points are resorted, and a 'bottom to top' scan is performed. There are several practical tolerancing considerations with this method, as the points are not on an even grid, and there may not be matching x or y coordinate values.

Once the boundary points are identified, the boundary curves that the points lie upon are labeled appropriately using a nearest point algorithm. The points that are 'near' each other (within a user defined limit) are determined to lie on the same curve. These points are used to derive a boundary curve. This methodology is a subset of the approach used to defined features from a point cloud for design recovery [20].

All the points and boundary curves can be generated for each orientation angle [phi] and then the common region determined via Boolean intersection routines. Alternatively, an 'in or out' comparator can be implemented to compare points from one orientation to another. 'Valid' points share the same space.

Both methods are computationally expensive, and the end results are impacted by the increment parameter [DELTA]. Hence, alternative methods are investigated.

Kinematic Synthesis Approach

With the kinematic synthesis approach, [[theta].sub.4] and [[theta].sub.6] are fixed, as is [[theta].sub.1] via radially projecting points on a curve or travel path onto 2D space. The forward kinematic relationships are used in a structured manner to determine the position of the essential end effector points via analysing the links and joint angle constraints of [[theta].sub.2], [[theta].sub.3], and [[theta].sub.5]. Geometric operations are paired with Boolean intersection routines to determine the functional workspace region for a set of end effector orientation [15]. The 2D projected travel path point set is superimposed onto this result to determine whether the reach condition criterion is met. The general methodology follows:

Derive the outer primary outer boundary radius [O.sub.Br]:

* Set [[theta].sub.1] to 0, and ignore link [d.sub.6] as this will be used with the end-effector to offset the functional work space region.

* Use the [[theta].sub.2] origin (center [O.sub.Bc] ) as the pivot point and link lengths [a.sub.2], [a.sub.3], and [d.sub.4].

Trim the outer boundary to right and left outer boundary circle centers [O.sub.Rc] and [OL.sub.c] respectively, which are located at the end of link [a.sub.2] or [a.sub.3].

* Set [[theta].sub.2] to its minimum value [[theta].sub.2min] and create an arc [OL.sub.c] with radius link 4, [d.sub.4]

* Set [[theta].sub.2] to its maximum value [[theta].sub.2max] and create an arc [OR.sub.c] with radius link 4, [d.sub.4]

* Trim the curves at their intersection points, and create one continuous curve (Figure 11)

Derive the outer primary inner boundary radius [I.sub.Br] using the Cosine Law. Two sides ([a.sub.2] + [a.sub.3]), and [d.sub.4], and the [[theta].sub.3] joint limit provide the information required. This is illustrated in Figure 12.

If the limits [[theta].sub.5max] and [[theta].sub.5min] are [+ or -] 180[degrees], then the boundary curves are just offset. In Figure 13, three offsets are applied for three different end effector orientations. Both the inner and outer boundary curves are offset. Boolean intersection routines are applied to generate a bounded region of reachable points. The outer boundary curve is determined, defining the region where all points can be reached for the end-effector orientation / orientation set. Here in Figure 14 the boundary is trimmed to [[theta].sub.1] pivot axis.

Including the inner boundaries, and re-trimming the model generates the boundary shown in Figure 15:

In most cases, the rotary limits [[theta].sub.5m] and [[theta].sub.5min] do not complete a full circle, and the limits need to be used to generate relevant boundaries. The [[theta].sub.5] rotary limits, in combination with the end effector orientation angle [phi] and end-effector length, are used to create a set of offset bounded regions. The joint five angle [6.sub.5] for data generation of valid points within the functional workspace is [16]:


Where [phi] is the end effector orientation,

K = cos([a.sub.2]), and

[a.sub.2] is the joint 2 twist angle.

From this, the maximum joint angle [[theta].sub.5] can be determined, the curves offset to these limits, and a new outer boundary curve derived. As with before, each end effector angle influences the outer boundary, but including the joint five limits further constrains the feasible region where all points can be reached (Figures 16 and 17).

Joint angle [[theta].sub.5] joint limits may have limits such that the [[theta].sub.2max] limit cannot be reached for a given end-effector orientation. Therefore, elbow up / elbow down solution must be assessed, or the trim regions reconsidered (i.e. do not use the joint 1 pivot axis as a trimming boundary. Note: disjoint regions may occur when considering these solution sets, and this information can also be leveraged to identify regions where no joint wrap (i.e., elbow up, elbow down) occurs.

D-H Parameter Based Solutions

Another solution approach incorporates employing the D-H parameters, the kinematic structure and link limits (Figure 18) to generate reachable points, and compare the regions [21]. This solution is developed using the MATLAB tool box. As a serial link manipulator is a series of links, which connects the end-effector to the base, each link is physically connected to the next by an actuated joint. If a coordinate frame is attached to each link, the relationship between two links can be described with a homogeneous transformation matrix using the D-H rules, and they are named [??], where i is number of joints, as shown in Equation (2).


The robot can now be kinematically modeled by using link transforms:



[.sup.0.A.sub.n] is the pose of the end-effector relative to base;

[.sup.i-1.A.sub.i] is the link transform for the ith joint; and

n is the number of links.

For the KUKA KR robot family, the six homogeneous transformation matrices have been developed. The pose matrix of the end-effector relative to base is presented in equ. (4). The end-effector orientation is defined with the rotation matrix in equ. (5).



The orientation of the end-effector, relative to the robot base frame, is defined with the three vectors: n, s, and a. The functional work window has been calculated using the kinematic equation (equation 6) and calculated joint five angular position [[theta].sub.5] in a function of joint two and three joint angles.


The functional reach regions have been generated for [[theta].sub.5] = -90[degrees] (270[degrees]), -45[degrees] (315[degrees]) and 135[degrees] (225[degrees]), and are shown in Figure 19. This methodology is applied to the Comau robot (Figures 20 and 21), except [[theta].sub.5] = 90[degrees] to illustrate disjoint regions. The boundary curves can be developed from this point set, or a point 'in or out' algorithm can be applied for the 'squashed' travel path points.


Laser cladding is a process where a laser heats an alloy powder or wire, and a bead of material is deposited onto a substrate. Only a thin layer of the substrate material is melted. The cladding operations can done to build up material on a worn component (i.e., mold refurbishment), or to place a protective layer of material on a surface of a component to improve corrosion or abrasion resistance [22, 23].

The travel path for mold refurbishment must take into consideration both the laser spot size and shape: in many cases, a square spot is employed. Hence specialized travel strategies must be developed while considering the system constraints. A robot is employed with a table-table positioning system [24] to ensure the torch angle [phi] is always perpendicular to the surface ( [phi] = -90[degrees]), but the surface normals are dealt with (Figure 22 (a)). Edging the outer contours requires a 5 axis tool path, hence the torch angle is rotated 45[degrees] for this path. Therefore, a [phi] = -45[degrees] (315[degrees] and [phi] = -135[degrees] (225[degrees]) constraint exists as well. For the surface shown in Figure 22 (b), the travel path, and a reduced point set are shown.

Placing the mold in the cell at a selected reference position, and projecting the points, it can be seen that there are no reach condition issues for either robot. Alternatively, this point set can be compared to the empirical or D-H parameter based point set to determine whether reach issues exist.

Downstream optimization rules can be employed using various robotics' simulation packages to improve the travel path.

Determining the functional working region or work window, and establishing the appropriate datum points for a desired travel path needs to be done prior to introducing higher level optimization strategies. The working boundary information presented by various manufacturers is appropriate for addressing layout and safety concerns, but the feasible working region varies based on the kinematic structure, tool parameters, and the required end effector and tool orientation, and is a subset of the work envelope. Forward kinematic based strategies are presented which reduce the problem from a 3D complex problem to a series of smaller 2D challenges using two constraint sets: (i) developing radial projections or squashing appropriately onto a reference plane, and (ii) determining 2D representations of the functional work space. This reduces the problem to an open chain four bar linkage variant. The radial projection strategy addresses the joint-link 1 impacts. The three methods to develop a functional working window all focus on assessing the joint-link 2, 3, and 5 impacts. The techniques are summarized as follows, and in Table 3:

* Empirical approach (data management challenges)

** Simple and intuitive approach

** Can be employed by someone with little in-depth knowledge of robotic kinematics or programming to generate a point set

** Slow (minutes to hours depending on the resolution and number of orientations), need to employ data collection, sorting, and curve fitting routines

* Geometric approach (uses minimal information with respect to the robot kinematic structure and joint constraints, multiple joint combinations need to be considered)

** Uses minimal information and standard geometry creation tools, which can be implemented either manually or algorithmically in standard CAD/CAM packages using a software development kit (SDK).

** Boolean intersections can be readily determined

** Each joint orientation combination needs to be assessed; consequently, care must be taken when performing intersection routines

** Results can be generated quickly using standard CAD/CAM software

* D-H parameter approach (uses minimal information with respect to the D-H parameters and joint constraints, multiple joint combinations are combined in the results)

** Uses the standard D-H parameters notation

** Readily implemented using the MATLAB toolbox

** Faster than the empirical approach, but there is still the need to employ data collection, sorting, and curve fitting routines

** Establish feasible regions which are then used for other downstream robotic MATLAB simulations

Each method can be employed to determine the joint reach feasibility prior to physical set ups or modifications. For all methods, the impact of a new travel path or end-effector orientation can be evaluated without sophisticated knowledge of robotic design and simulation tools. The empirical solution approach can be used for teaching, or error checking / debugging the other methods, or for assessing unique configurations. The DH parameter-MATLAB based method is automated, and can be linked to additional MATLAB robotic / optimization modules. However, MATLAB is typically not employed in manufacturing settings by support personnel. The geometric solution is a fast method and can be expanded to visualize 3D layout solutions, as revolved volumes and overlap regions for multiple kinematic chains in a work cell can be readily generated. To conclude, three methods are presented to evaluate the feasible work region. This needs to be determined prior to downstream optimization analyses with respect to work cell process planning and layout, as well as the tool path generation and optimization. The circumstances will dictate which methodology is most appropriate.


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Jill R. Urbanic

Phone: (519)-253-3000 ex. 2633


[.sup.0.A.sub.n] - End-effector frame with respect to the base frame

[.sup.i-1.A.sub.i] - Frame transform matrix of the ith joint with respect to i-1

n - Number of links

[a.sub.i] - Link length is the distance along the common normal between the joint axes

[d.sub.i] - Link offset is the displacement, along the joint axes between the links

[[ALPHA].sub.i] - Twist angle - the angle between the joint axes

[0.sub.i] - Joint angle - the angle between the links

[empty set] - End effector orientation

[DELTA] - Increment parameter

[O.sub.Rc] - Right outer boundary circle center

[O.sub.Lc] - Left outer boundary circle center

[O.sub.Br] - Outer boundary circle center

[I.sub.Br] - Inner boundary radius

R.J. Urbanic University of Windsor R. Hedrick CAMufacturing Solutions Inc. Ana M. Djuric Wayne State University

Table 1. The D-H parameters for the KUKA KR robot

              D-H parameters
i  [d.sub.i]  ot              [[omega].sub.i]  [[omega].sub.i]

1  [d.sub.1]  180[degrees]    [a.sub.1]  -90[degrees]
2  0           90[degrees]    [a.sub.2]    0[degrees]
3  0            0[degrees]    [a.sub.3]  -90[degrees]
4  [d.sub.4]    0[degrees]    0           90[degrees]
5  0            0[degrees]    0           90[degrees]
6  [d.sub.6]  180[degrees]    0          180[degrees]

    Joint         Limits
i   Minimum       Maximum

1  -185[degrees]  185[degrees]
2   -40[degrees]   93[degrees]
3  -210[degrees]   58[degrees]
4  -350[degrees]  350[degrees]
5  -120[degrees]  120[degrees]
6  -350[degrees]  350[degrees]

Table 2. The D-H parameters for the Comau robot

              D-H parameters
i  [d.sub.i]  [theta].sub.i]  [[omega].sub.i]     [[omega].sub.i]

1  l          [d.sub.1]        90[degrees]  [a.sub.1]
2  2          0                90[degrees]  [a.sub.2]
3  3          0               180[degrees]  [a.sub.3]
4  4          [d.sub.4]       180[degrees]  0
5  5          0                 0[degrees]  0
6  6          [d.sub.i]       180[degrees]  0

   Joint         Limits
i  Minimum       Maximum

1  -90[degrees]  -171[degrees]
2  180[degrees]   -55[degrees]
3   90[degrees]  -162[degrees]
4  -90[degrees]  -280[degrees]
5   90[degrees]  -120[degrees]
6  180[degrees]  -270[degrees]

Table 3. Solution summary (0 no expertise, 1 lowest, 3 highest)

                    Process Flow             Input

                    Develop a kinematic
                    Manually develop a
                    2D set of points         Linkage data:
                    * Select an increment    * link
                    angle                    lengths,
                    * Increment joint        * joint
                    angles                   constraints
                    Extract boundary         Increment
                    points                   angle
                    * Sort the points        End effector
                    * Apply IN-OUT filter    orientation set
                    Develop bounding
                    curves using NURBs
                    modelling tools

                    Develop a kinematic      Linkage data:
GEOMERICAL          model                    * link
                    Determine bounding       lengths,
                    geometry                 * joint
                    Boolean operations to    constraints
                    develop the bounded      End effector
                    region                   orientation set

                    Develop a kinematic
                    Automatically            DH
                    develop a 2D set of      Parmenant Parameters
de parameter based  points                   Linkage data:
                    * Select an increment    * link
ETER                angle * Increment joint  lengths,
                    angles                   * joint
                    Extract boundary         constraints Increment
                    * Sort the points        angle End effector
                    * Apply IN-OUT filter
                    Develop bounding         orientation set
                    curves using NURBs
                    modelling tools

                    Process  Tools         Prog.
                    Time     Required      Expertise

                                           0 to
                                           generate the
EMRIRICAL                    CAD           point set

                    3        or robotic    3
                             simulation    automate

                             software      boundary generation

                             CAD           0-
                             modelling     manually
                             tools, +      create
GEOMERICAL                   SDK for       boundaries
                             automating    using CAD
                    1        the solution  commands

                             SDK           3
                             program-      automate
                             ming          boundary
                             knowledge     generation

de parameter based                         3 to
                                           generate the
ETER                2        MATLAB        point set
                             tool box      3
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Author:Urbanic, R.J.; Hedrick, R.; Djuric, Ana M.
Publication:SAE International Journal of Materials and Manufacturing
Article Type:Report
Date:May 1, 2016
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