This paper examines the problem of geometric constraints acquisition of planar motion through a line-geometric approach. In previous work, we have investigated the problem of identifying point-geometric constraints associated with a motion task which is given in a parametric or discrete form. In this paper, we seek to extend the point-centric approach to the line-centric approach. The extracted geometric constraints can be used directly for determining the type and dimensions of a physical device such as mechanical linkage that generates this constrained motion task.

## Introduction

This paper studies the problem of planar motion generation from the perspective of line-geometric constraint identification and acquisition, following our previous work [1] that advocated a point-geometric approach to the same problem. Kinematic acquisition of geometric constraints is concerned with the identification or extraction of geometric constraints that are embedded in an explicitly given motion, which is defined either parametrically or discretely as an ordered sequence of displacements. The resulting geometric constraints can be used to obtain an implicit representation of the same motion, approximately. This process is called approximate implicitation of a motion.

This paper advocates a geometric-constraint based approach with a focus on the analysis of line trajectories associated with the motion, the goal of which is to obtain a line trajectory that can be constructed as a geometric condition that best describes the motion. Typically, this is done in a geometric constraint identification and acquisition process, i.e., by comparing various trajectories of a specified motion with known constraints from a library of mechanically realizable constraints. This effectively reduces the problem of mechanism synthesis to that of constraint identification and acquisition, and thus bridges the gap between type and dimensional synthesis.

The organization of the paper is as follows. Section 2 reviews the concepts of line trajectory and envelope curve for the development of this paper. Section 3 discusses the equiform transformation for line constraints. Sections 4 and 5 deal with the acquisition of geometric constraints in explicit form and implicit form, respectively. Section 6 illustrates with examples to demonstrate the effectiveness of the proposed approach.

## Preliminaries on Line Trajectories

This section provides a review of line trajectories in a plane insofar as necessary for the development of this paper.

### Line Transformation.

Consider a planar displacement a rigid body shown in Fig. 1, which is composed of the translation (dx, dy) of a point on the moving body and the orientation angle α of the moving body. A coordinate frame denoted M is attached to the moving body, while F represents the fixed coordinate frame. It is typical to represent the planar displacement by a point coordinates transformation from M to F

$V=[H(dx,dy,α)]v=[ cos α−sin αdx sin α cos αdy001]v$
(1)

where $v=(v1,v2,v3)$ and $V=(V1,V2,V3)$ are homogenous coordinates of a point in F and M, respectively.

One may also represent the same planar displacement in terms of line coordinates transformation. Let $l=(l1,l2,l3)$ and $L=(L1,L2,L3)$ denote the homogenous coordinates of a line in F and M, respectively. Then, the line transformation is given by Bottema and Roth [2] as follows:
$L=[H¯(d¯x,d¯y,α)]l=[ cos α−sin α0 sin α cos α0d¯xd¯y1]l$
(2)
where $d¯x=−dx cos α−dy sin α, d¯y=dx sin α−dy cos α.$ As shown in Ref. [2], the point and line transformations satisfy the following relation:
$[H¯(d¯x,d¯y,α)]T[H(dx,dy,α)]=[I]$
(3)

where [I] is the 3 × 3 identity matrix.

### Envelope of a Line Trajectory.

When the parameters $(d¯x,d¯y,α)$ are functions of a parameter t, a line $l$ in the moving body traces out a family of lines $L(t)=[H¯(dx(t),dy(t),α(t))]l$ called line trajectory of the moving line. Now consider the envelope curve traced by the line trajectory. The line envelope can be interpreted as the collection of the intersections of nearby lines $L(t)$ at all time t. The intersection at certain time t is the point at which the line tangentially touches the envelope at that instant. Therefore, the point and line must satisfy the enveloping condition
$V1dL1(t)dt+V2dL2(t)dt+V3dL3(t)dt=0$
(4)
In addition, the line passes the intersection point, which indicates the following incidence relation:
$V·L(t)=V1L1(t)+V2L2(t)+V3L3(t)=0$
(5)

Details on the general theory of envelopes from the perspective of differential geometry can be found in Refs. [10,11]. Line geometry has also found applications in computer-aided design of ruled surfaces and line congruences [1216]. The connection of points and lines has also been studied by Zhang and Ting [17].

### Projective Duality.

Now consider the following linear equation:
$V1L1+V2L2+V3L3=0$
(6)

This equation can be either interpreted as constraining a point V on a line L or as constraining a line L such that it passes through a point V. The role of point coordinates V and line coordinates L is completely symmetric. This gives rise to the principle of duality in the projective plane, i.e., geometric transformations that replace points by lines and lines by points while preserving incidence properties among the transformed objects. In other words, any theorem or statement that is true for the projective plane can be reworded by substituting points for lines and lines for points, and the resulting statement will be true as well [18].

Consider the problem of computing the tangent line $L(t)$ for the envelope curve $V(t)$. Using the principle of duality, we may switch the role between points and lines to consider $V(t)$ as representing moving lines $Ṽ(t)$ and $L(t)$ as representing points $L̃(t)$. Thus, the problem can be converted to that of computing the envelope curve $L̃(t)$ of the tangent line $Ṽ(t)$. In this case, Eqs. (4) and (5) become
$L̃1dṼ1(t)dt+L̃2dṼ2(t)dt+L̃3dṼ3(t)dt=0$
(7)

$L̃·Ṽ(t)=L̃1Ṽ1(t)+L̃2Ṽ2(t)+L̃3Ṽ3(t)=0$
(8)

In conclusion, the mutual generation between the line motion and the envelope curve is bijective and unambiguous in the sense that one line motion results a unique envelope curve and this curve recovers the same lime motion. The converse statement is also true: a curve can set up a unique line motion and this line motion duplicates the same curve as its envelope. Consequently, the line trajectory and the envelope curve are dual to each other; the representation of a line trajectory can be replaced by its envelope curve and vice versa.

## Equiform Transformation of Geometric Constraints

In our recent paper [1], we have employed an equiform transformation that allows for rotation, translation, and uniform scaling of a point-geometric shape in the plane
$[E]=[a−bcxbacy001]$
(9)

where $a=λ cos ξ$ and $b=λ sin ξ$ with ξ being the angle of rotation and λ being the scaling factor. The translation component is given by (cx, cy). The matrix $[E]$ contains four independent parameters $(a,b,cx,cy)$.

In terms of line coordinates, the equiform transform is given by
$[E¯]=[ cos ξ−sin ξ0 sin ξ cos ξ0D¯xD¯yκ]$
(10)

where $D¯x=−cx cos ξ−cy sin ξ, D¯y=cx sin ξ−cy cos ξ$, and $κ=1/λ$. Thus, the line-geometric equiform transform has also four degrees of freedom and are defined by the parameters $(ξ,κ,D¯x,D¯y)$.

For a given motion, the main objective of kinematic acquisition of line-geometric constraints is to identify a line l in the moving body such that its trajectory $L$ best approximates a desired line trajectory $G$, as shown in Fig. 2. $G$ is equiformly transformed from the standard line constraint $g$, i.e., $G=[E¯]g$. Instead of being directly given as a line trajectory, the standard line constraint can also be indirectly given as its corresponding envelope. In this case, Eqs. (7) and (8) are used to convert the envelope constraint to the line constraint.

In the matching process, one notable factor is parameterizations of the constraint curve and the trajectory L. Ideally, an intrinsic parameterization $L(t)$ for the line trajectories can be carried out to remove the influence of parameterization in the shape matching process. For example, one would like to convert the line trajectories to the envelope curves, hence the arc length for each of the envelopes can be carried out to remove the influence of parameterization in the shape matching process, and thereafter, computational comparison of either the envelopes or the envelope arc-length parameterized line trajectory leads to a solution to the constraint retrieval. However, in this paper, we take a computationally less expensive approach of normalizing the parameterizations of these line trajectories so that they are in the range of $[0,1]$. Since the periods of these trajectories are made equal to one, we have
$g(t)=g(t−1),G(t)=G(t−1),L(t)=L(t−1)$
(11)
where $t=∈[0,1]$. Another factor that affects the constraint matching process is the choice of initial line on each of these trajectories. To address this issue, we include a shift factor $tΔ$ so that the initial line on the constraint line trajectory can vary in order to find a better match. Incorporating the shift factor, we can define the normalized parameters, tG and tL, for the constraint curve $G(tG)$ and a point trajectory of the given motion, $L(tL)$, as
$tL=t, tG=t+tΔ$
(12)

wherein $t∈[0,1]$ and the shift parameter $tΔ$ indicates how much the motion-shape correspondence has to be shifted. As a result, $L(tL)$ on $[0,1]$ match $G(tG)$ on $[tΔ,1+tΔ]$.

## Acquisition of Geometric Constraints in Explicit Form

This section presents an algorithm for identifying a given constraint from line trajectories of a specified motion task. This work can be considered as an extension of our earlier work on kinematic acquisition of point trajectories [1].

We assume that the motion task has been given explicitly as an ordered sequence of N discrete positions and a set of N ordered lines $gi=(cos ηi, sin ηi,gi)$ belonging to a specified line constraint has also been given. Consequently, instead of using continuous parameters tL and tG as given by Eq. (12), we use indexes i ($i=1,…,N$) to represent the sequence of lines and positions, where the shift parameter $tΔ$ becomes an integer n, which means that the starting line on the geometric constraint line set is shifted by n lines. Thus, the squared distance between the ith line on the trajectory L and the $(i+n)$ th line on the geometric constraint is given by
$δi,n2=(Li−Gi+n)T(Li−Gi+n)$
(13)
In view of Eqs. (2) and (9), we have
$Li=[H¯i]l, Gi+n=[E¯]gi+n$
(14)
where $[H¯i]$ is obtained from the 3 × 3 matrix in Eq. (2) by replacing $α,d¯x,d¯y$ with $αi,d¯x,i,d¯y,i$, respectively. Here, we assume that $l$ is normalized such that $l=(cos β, sin β,l)$, where $(cos β, sin β)$ is the unit vector perpendicular to the line and pointing from the origin of moving coordinate frame to the line, and l gives the perpendicular distance from the origin of moving coordinate frame to the line. Substituting Eq. (14) into Eq. (13), after some algebra, we obtain
$δi,n2=(Ai,nTX−[d¯x,i,d¯y,i][cos β, sin β]T)2+2−2 cos(β+αi−ξ−ηi+n)$
(15)
where
$Ai,nT=[cos ηi+n, sin ηi+n,gi+n,−1]$
(16)
are determined from the specified motion and the geometric constraint to be matched, and
$X=[D¯x,D¯y,κ,l]T$
(17)
are the design variables associated with the equiform displacement and choice of the line $l=(cos β, sin β,l)$ on the moving body. The average of squares of the deviations is thus given by Eq. (15)
$S=1N∑i=1Nδi,n2=1N∑i=1N{(Ai,nTX−[d¯x,i,d¯y,i][cos β, sin β]T)2+2−2 cos(β+αi−ξ−ηi+n)}$
(18)

The goal for constraint acquisition is to find a line $l=(cos β, cos β,l)$ on the moving body such that the trajectory of l best approximates the given line trajectory constraint after an appropriate equiform displacement, $(ξ,κ,D¯x,D¯y)$. So in total, we have six variables, $(β,l,ξ,κ,D¯x,D¯y)$, to be determined.

The least square solution to Eq. (18) satisfies the following conditions:
$∂S∂D¯x=0, ∂S∂D¯y=0, ∂S∂κ=0, ∂S∂l=0, ∂S∂ξ=0, ∂S∂β=0$
(19)
The first four conditions give
$[∑i=1N{Ai,nAi,nT}]X=[∑i=1N{Ai,n[dx,i,dy,i]}][ cos β sin β]$
(20)
which allows X to be expressed in terms of α. The fifth condition leads to
$∑i=1N sin (β+αi−ξ−ηi+n)=0$
(21)
which contains only β and ξ. The last condition yields
$[cos β, sin β]([∑i=1N{[dy,i−dx,i]Ai,nT}]X−[∑i=1N{[dy,i−dx,i][dx,i,dy,i]}][ cos β sin β])+∑i=1N sin (β+αi−ξ−ηi+n)=0$
(22)
The above equation can be simplified by substituting Eqs. (20) and (21) into it. After some algebra, it can be reduced to
$[cos β, sin β]M[cos β, sin β]T=0$
(23)
where
$M=[∑i=1N{[d¯y,i−d¯x,i]Ai,nT}][∑i=1N{Ai,nAi,nT}]T[∑i=1N{Ai,n[d¯x,i,d¯y,i]}]−[∑i=1N{[d¯y,i−d¯x,i][d¯x,i,d¯y,i]}]$
(24)

Equation (23) contains only one unknown β, which can be solved using tangent half-angle substitution. Substituting β into Eq. (21), we obtain a trigonometric equation with one unknown ξ, which again can be obtained using tangent half-angle substitution. Finally, substituting β and ξ into Eq. (20), we obtain a 4 × 4 linear equation with unknown X, which can be readily solved.

So far, it has been assumed that the shift parameter n is given. In practice, the desired value of n is unknown beforehand. We now present a numerical algorithm for shape matching which combines least squares optimization with a direct search to deal with the shift parameter n:

Objective: Given a discrete motion M with N sampled positions and a geometric constraint line trajectory $g$ of N sampled points, find a line on the moving body $l=(cos β, sin β,l)$, such that after the constraint line trajectory $g$ is transformed into to a new configuration $G$ via an equiform displacement, $(ξ,κ,D¯x,D¯y)$, the error between the trajectory L and the constraint line trajectory $G$ as defined by Eq. (18) is at minimum.

Algorithm

For each $n∈[1,…,N]$do

For each $i∈[1,…,N]$do

$•$ Initialize values of $Ai,n,d¯x,i,d¯y,i$;

$•$ Substitute $Ai,n,d¯x,i,d¯y,i$ into Eqs. (20), (21), and (23);

End for

$•$ Solve Eq. (23) for β;

$•$ Substitute β into Eq. (21) and solve for ξ;

$•$ Substitute β and ξ into Eq. (20) and solve for X;

If the total error is currently at minimum do

$•$ Update final solution to ${β,l,ξ,κ,D¯x,D¯y,n}$;

End if

End for

## Acquisition of Geometric Constraints in Implicit Form

The algorithm presented in Sec. 4 assumes that the line geometric constraints be explicitly given. This section studies the case where the constraints are implicitly given. We restrict our discussions to conics as they can be readily generated using simple 2DOF mechanisms (Fig. 3). A comprehensive treatise on the generation of the tangents to planar algebraic curves using planar mechanisms can be found in Ref. [19].

The envelope curves of those line constraints generated by mechanisms shown in Fig. 3 are conic curves, which can be implicitly defined by the following quadric equation:
$aVx2+bVxVy+cVy2+dVx+eVx+f=0$
(25)
When the quadric is not irreducible, the conic is said to be nondegenerate. Such conics include circles, ellipses, hyperbolas, and parabolas. If the quadric factors into a product of linear polynomials, then the conic is just the union of two lines and said to be degenerate. A conic curve can be expressed in matrix form as
$VT[C]V=0 with [C]=[ab/2d/2b/2ce/2d/2e/2f]$
(26)

where [C] is called the matrix of the conic section, and nonsingular for nondegenerate conics. A conic is homogenously described by the six coefficients $(a,b,c,d,e,f)$.

The tangent to the conic at point V is given by
$L=[C]V$
(27)
Let $[C*]$ denote the adjoint matrix of [C]. Then, it is easy to verify that the tangent line L given by Eq. (27) satisfies the following condition:
$LT[C*]L=0$
(28)

This means that the coordinates of the tangent line also satisfy a quadric equation, which is said to define a conic. Due to the principle of duality, the new conic $[C*]$ is said to be dual to the original conic [C].

Let $[E(ξ,λ,cx,cy)]$ denote an equiform transform. After the transform, the new conic is given by
$[C¯]=[E]−T[C][E]−1$
(29)
and the dual conic for the tangent lines is
$[C¯*]=[E][C*][E]T$
(30)
All the tangent lines of new configuration satisfy
$LT[C¯*]L=0$
(31)
Again, the goal of kinematic acquisition is to identify a line in the moving body as well as the equiform transform parameters $(ξ,λ,cx,cy)$ such that the envelope of the line trajectory L best approximates the equiformly transformed configuration $[C¯*]$ of a desired conic [C]. When the motion task is given explicitly as an ordered sequence of N discrete positions, the approximation error for the conic constraint is given by
$S=∑i=1N(LiT[C¯*]Li)2$
(32)
Substituting $Li=[H¯i]l$ and Eq. (30) into the above equation, we have
$S=∑i=1N(lT[H¯i]T[E][C*][E]T][H¯i]l)2$
(33)

The goal is to find $β,l,ξ,λ,cx,cy$ to minimize the above error. In this case, the error function is a quadric polynomial of $cos β, sin β,l, cos ξ, sin ξ,cx,cy$, and λ. This means that we cannot solve them explicitly as we have done in Sec. 4. In this paper, a numerical method called simulated annealing is used to search for the solutions.

## Example

In what follows, we present an example to illustrate the effectiveness of our constraint identification and acquisition scheme. The given motion task is artificially designed such that one of the lines (0, 1, 0) in the moving body remains tangent to an ellipse, which has a major axis of length 5, a minor axis of length 3, and its center is located at point $(−1.5,1.0)$. The angle between the major axis of the ellipse and the horizontal axis of the reference frame is 30 deg.

In the first part of this example, we test the algorithm presented in Sec. 4 that assumes that the tangent lines of the constraint curve are explicitly given. We use the green ellipse in Fig. 4 as our desired shape. For explicit constraint, the ellipse is given as a sequence of sample points. For the purpose of testing the algorithm, sample points on the constraint are identical to those on the ellipse used to construct the example motion. Hence, we expect the line (0, 1, 0) to be identified as a tangent moving line to the ellipse. Figure 5 depicts exactly the same constraint as the given one, while Fig. 6 shows another identification result which is also an elliptical constraint. The types and dimensions of the mechanisms producing the two identified constraints are also shown in Figs. 5 and 6. Both mechanisms are capable of generating the task motion independently.

Both synthesized mechanisms are of 2DOF and same type as the five-link mechanism shown in Fig. 3(b) from Sec. 5. A line on its end-effector moving frame is always tangent to an ellipse, as specifically demonstrated in Fig. 7. The fixed pivot A and C coincides with the center and one of the foci of the ellipse, respectively. The length of the crank AB is equal to half the major axis of the ellipse, while the distance between the axes of fixed pivot A and C is equal to half the focal distance. The end-effector moving frame is rigidly connected with the slide-block, located at the prismatic joint D. A line on the moving frame and collinear with the link PD remains always tangent to the ellipse at the point P. For the proof that the curve enveloped by the line PD is actually an ellipse, see Artobolevsky [19] for details. Thus, two elliptical constraints are identified by our line geometric approach, with the types and dimensions of the associated mechanisms being simultaneously determined.

In the second part of the example, we examine our kinematic acquisition algorithm for the implicitly specified constraint as presented in Sec. 5. Also, the elliptical constraint is used for testing and given by the following matrix of the conic section:
$[C]=[12.5200011.52000−1]$
(34)

In this case, in addition to the original constraint, we have obtained multiple solutions of good quality, as demonstrated in Figs. 8 and 9. The two 2DOF synthesized mechanisms, associated with the identified constraints and as well shown in Figs. 8 and 9, are of the same type but different dimensions, either of which can be selected to generate the task motion. Furthermore, the task motion can be viewed as a planar motion subject to the two identified constraints. Therefore, a 1DOF closed-loop mechanism can be synthesized, as shown in Fig. 10, with the two 2DOF mechanisms as components.

## Conclusions

In this paper, we have successfully extended the method to extract point geometric constraints from a specified motion task to line geometric constraint by resorting to the duality of point and lines. The resulting constraints can then be used to identify and synthesize simple mechanisms for constraint generation. This task-oriented and constraint-based paradigm to mechanism design greatly reduces the complexity of the simultaneous type and dimensional synthesis problem.

## Acknowledgment

The work has been financially supported by U.S. National Science Foundation (Grant No. CMMI-1563413) and the Fundamental Research Funds for the Central Universities of China (No. 2682015BR004). The authors also thank Sichuan Provincial Machinery Research and Design Institute (China) for their support. All findings and results presented in this paper are those of the authors and do not represent those of the funding agencies.

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