On the Configuration Space of planar closed Kinematic Chains
Gerhard Zangerl



Multiscale Fluctuation-based Dispersion Entropy and its Applications to Neurological Diseases
    Hamed Azami^1,*, Steven E. Arnold^2, Saeid Sanei^3, Senior Member, IEEE, Zhuoqing Chang^4, Guillermo Sapiro^5, Fellow, IEEE, Javier Escudero^6, Member, IEEE, and Anoopum S. Gupta^7
^1Department of Neurology and Massachusetts General Hospital, Harvard University, Charlestown, MA 02129, USA.
	
*Corresponding author, email: hazami@mgh.harvard.edu.
^2S. E. Arnold is with the Department of Neurology and Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129, USA.
^3S. Sanei is with the School of Science and Technology, Nottingham Trent University, Nottingham, UK.
^4Z. Chang is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27707, USA.
^5G. Sapiro is with the Department of Electrical and Computer Engineering, Computer Sciences, Biomedical Engineering, and Math, Duke University, Durham, NC 27707, USA.
^6J. Escudero is with the Institute for Digital Communications, School of Engineering, University of Edinburgh, Edinburgh, UK.
^7A. S. Gupta is with the Department of Neurology and Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA.
    December 30, 2023
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A planar kinematic chain consists of n links connected by joints. In this work we investigate the space of configurations, described in terms of joint angles, that guarantee that the kinematic chain is closed. We give explicit formulas expressing the joint angles that guarantee closedness by a new set of parameters, the diagonal lengths  (the distances of the joints from the origin) of the closed kinematic chain. Moreover, it turns out that these diagonals are contained in a domain that possesses a simple structure. We expect that the new insight can be applied for several issues such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. In order to demonstrate practicality of the new method we present numerical examples.   











§ INTRODUCTION








In this work we investigate  the configuration space of closed planar kinematic chain (CKC) with n links connected by revolute joints in terms of its joint angles. In many fields like robotics computational biology or protein kinematics it is of immense interest to understand the configuration space of a CKC. For instance, in robotics the problem to connect a start, α_s, and goal configuration, α_g naturally appears and thus requires knowledge of the configuration space, which is typically a manifold or variety in the ambient space formed by the robots joint variables. The configuration space is even more complicated if additional constraints like obstacle, link-link avoidance, or limited joint angles are included. Two main strategies, probabilistic and geometric approaches, to investigate configuration spaces have been developed so far.



Probabilistic methods have been successfully applied for constrained motion planning. They are especially important in practical situations with high dimensions that include complex constraints such as obstacle avoiding. Typically these methods are based on the generation of random configurations in ambient joint space followed by a check up if they approximately satisfy the desired constrains. Repeating this procedure results in a discrete version of the configuration space that is very useful in applications. Probabilistic  methods have been applied in different situations, which can be found in 
<cit.>,



Besides the approaches using randomness other works focused on questions about the geometry and topology of the  configuration spaces of kinematic chains. Insight about the global geometry of configuration spaces is very important in applications. Early discoveries have been made by <cit.>. In their fundamental work Kapovitch and Milgram established important results about the geometry, which led to novel path planning algorithms. For instance in <cit.> it is used that the configuration space of a CKC consists of two connected components when it possess three long links. An application of this result is that path planning can be done easily for this special kind of CKC's. Also for the more difficult case, when CKC's do not have three long links algorithms were derived in <cit.>. They also developed path planners in the case of p point obstacles in the plane <cit.>. Another geometric approach was recently recognized by  Han, Rudolph and Blumenthal. They discovered that it is very beneficial to describe the configuration space of CKC by different parameters than the joint angles, see <cit.>. Their idea is to use the length of diagonals from the positions of revolute joints to the origin O as depicted on the right side of Figure <ref>. It turns out that for a CKC the length of these diagonals can be computed as solution of a system of linear inequalities, which means that all feasible diagonal lengths can be described by a convex polyhedron that can be handled by methods of linear programming <cit.>. Given feasible diagonal lengths, several configurations of the CKC can be constructed, since each link of the chain can be flipped over a diagonal. Thus in <cit.> any configuration can be obtained from a set of diagonals and a vector that represents the choices of flipping, which shows that the configuration space is formed by several copies of the polyhedron given by the system of inequalities. This practically convex structure is very useful for motion planning. In <cit.> paths between CKC with 1000 links are computed very efficiently.
        




Contribution of this work: We develop a new method that explicitly computes configurations of a CKC with n links, which are described by its joint angles. Compared to other methods it does not require linear programming to solve a system of linear inequalities like in <cit.> nor does it rely on probabilistic principles. The developed  method can be used to easily sample configuration space of a CKC and thus is expected to be useful in practical applications.
 


Outline of this text: In section <ref> we give a mathematical description of a CKC and its configuration space in terms of the diagonals of a CKC. Then the basic algorithm that explicitly describes how configurations of a CKC can be computed from its diagonals is developed in section <ref>. In section <ref> we describe the set of new parameters and show how they can be used to compute a vector of joint angles of  a CKC. Finally,  we give  numerical examples that show the validity of the developed method.     




§ CONFIGURATION SPACE



To describe the configuration space of a CKC with link lengths a_1, …, a_n, given as the entries of its vector of link lengths a^n= (a_1, …, a_n),  we introduce Cartesian coordinates in two dimensional Euclidean space. Moreover we place one of the links of the CKC so that it is supported by the positive x-axis and so that one of its ends coincides with the origin. Without loss of generality we can assume that the link a_n of the chain is fixed in the described manner, see Figure <ref>. In the following, we identify an angle α with its corresponding point on S^1. Furthermore, for 1 ≤ k ≤ n and a vector of angles α^k := (α_1, …, α_k)∈(S^1)^k we denote by 

    f_a^n ,k( S^1)^k →^2,     f_a^n ,k (α^k) = ∑_j=1^k a_j [ cos(α_j); sin(α_j) ].

the k-th endpoint map of a kinematic chain (KC). Additionally, the domain W_k :=  f_a^n ,k( (S^1)^k ), which is a circular annulus, will be referred to as the workspace of k-th endpoint map. 
We will call α^n-1 a configuration of the CKC with link lengths a_1, …,a_n if it satisfies the closure condition, which means that it is contained in the set

    (a^n) = {α^n-1∈(S^1)^n-1  f_a^n,n-1( α^n-1) =
    [ a_n;   0 ]} = f_a^n , n-1^-1(a_n, 0).

If no restrictions on the endpoint map are imposed α^n-1 will just be called a configuration of the KC with n-1 links.  

Furthermore,  the analysis in this work  uses the simple observation that it is sufficient to understand the space 
    (a^n)  =  {β^n-1∈( S^1)^n-1 f_a^n,n-1(β^n-1) ^2_2  = a_n^2 },

in order to describe (a^n), where ·_2 denotes the Euclidean norm. From the definition of (a^n) it is clear that any configuration β^n-1∈(a^n) satisfies that its endpoint 

f_a^n,n-1( β^n-1) ∈K_a_n

lies on the circle K_a_n that is centred on the origin and has radius a_n. We will say that β^n-1 is closed up to a rotation and call it a circular configuration of a CKC. 
Clearly, any circular configuration β^n-1 can be rotated by an angle λ,   

β^n-1 + λ:= ( β_1 + λ, …, β_n-1 +λ),  
 
so that β^n-1 + λ∈ C_a. Thus, if we are able to give an efficient method  to compute the set of solutions to the implicit equation   

 f_a^n, n-1(β^n-1	) ^2_2  = a_n^2, 

we also obtain configurations in (a^n) by the following two step algorithm:


  (i)   Compute a circular configuration  β^n-1∈(a^n)

  (ii)  Determine λ such that  α^n-1 =  β^n-1 + λ∈(a^n) 

Once a circular configuration is obtained step (ii) is a rather simple task. Therefore, in the following we will focus on the solution of step (i). This step is based on the fact that the trigonometric equation (<ref>), which in its expanded form is given as    

    ∑_i=1^n-1 a_i^2 + 2 ∑_i<j^n-1 a_i a_j 
    cos(β_i -β_j) = a_n^2 ,

allows for some kind of backwards substitution, see section <ref>. By the preimage theorem we know that the set of all circular configurations of a CKC with n links satisfying  (<ref>) is a manifold of dimension n-2, whenever a_n^2 is a regular value of the map g (β^n-1)  :=  f_a, n-1( β^n-1)    ^2_2. In all other cases the space _a may have singular points.




 §.§ Mathematical tools and notations



Surprisingly, the trigonometric equation (<ref>) can be rearranged into an equation of the same type but with one joint angle less appearing on its left hand side. For the computations we use  that a linear combination of sine and cosine functions can be written as      

    a sin( x ) + b cos( x) = c sin(x + φ(a,b)  ),

where c = √(a^2 + b^2)  and  φ(a,b)  = (a, b)
is the function described in Figure <ref>.

In order to  achieve a compact presentation of the results that will follow it is important to introduce abbreviations. For this purpose consider   

    2a_n-1∑_j=1^n-2 a_j cos(β_n-1 - β_j  ) + 2 ∑_i<j^n-2 a_i a_j  cos(β_i -β_j) + ∑_i=1^n-1 a_i^2 =  a_n^2

which is an equivalent form of (<ref>) that is obtained by fixing an index to be n-1 and rearranging remaining terms. Finally, using trigonometric summation formulas we arrive at    

    2 a_n-1sin( β_n-1) ∑_j=1^n-2 a_j sin( β_j )     + 2a_n-1cos( β_n-1) ∑_j=1^n-2 a_j cos( β_j ) +  ∑_i=1^n-1 a_i^2   
         +2∑_i<j^n-2 a_i a_j cos( β_i - β_j ) =  a_n^2.

In the last expression addition formula (<ref>) can be applied, which motivates the following abbreviations:


For a CKC with link lengths vector a^n = (a_1, a_2, …, a_n) for 1≤ k ≤ n-1 we set β^k := (β_1, …, β_k) and we abbreviate the x- and y-coordinates of the k-th endpoint map f_a^n,k by    

X_k(β^k):=  ∑_i=1^k a_i cos(β_i)   and    Y_k(β^k) :=  ∑_i=1^k a_i sin(β_i).

Using this notation we define 

L_k(β^k ) 	   := √(X_k(β^k)^2  + Y_k(β^k)^2  )  =  √( ∑_i=1^k a_i^2 +     2 ∑_i< j^k a_i a_j cos(β_i - β_j)),

which we refer to as the k-th diagonal length that  by definition can also be written as  L_k(  β^k ) =  || f_a^n, k(β^k)||. Moreover, consider the angle  
Φ_k (β^k)	   := φ(  X_k( β^k ),  Y_k( β^k )     ), 
which naturally appears, when applying formula (<ref>) for equation (<ref>). We set β^1 := β_1. We point out here that L_1( β^1 ) = a_1, L_n-1( β^n-1) = a_n. Additionally, we denote by L^n-3 = (L_2, … L_n-2) ∈^n-3 the vector of (variable) diagonals for a KC. Moreover, we use the notation 

    R_k^min   := 0 ∨max_1≤ i≤ k(2a_i-∑_j=1^ka_j)  and     R_k^max :=      ∑_j=1^ka_j    for 1≤ i≤ k,
 
where 0 ∨ a := max{ 0, a } for a∈. These quantities denote the outer and inner radius of the workspace W_k of the KC with links a^k = (a_1, …, a^k). Note that in computations carried out in section <ref> arguments of the defined quantities will frequently be omitted for simplicity and that with the notations equation (<ref>) can be rewritten as

    2 a_n-1sin( β_n-1) Y_n-2( β^n-2)     + 2a_n-1cos( β_n-1) X_n-2( β^n-2) +  L_n-2^2 =  L_n-1^2 - a_n-1^2.




 §.§ Computation  of Circular Configurations
 

We give a new method to obtain solutions to equation (<ref>). In the proof of Theorem <ref> a procedure how to obtain such solutions by reducing the length of the CKC step by step is described. For k ≥ 1 the solution method involves the choice of a real value in a domain that is given by an inequality that involves the diagonals of the CKC. Before we state the main theorem we give the following definition.      

 For a KC with a vector of links  a^n= ( a_1,…, a_n) and recall that L^n-3∈ℝ^n-3_≥ 0. We denote the diagonal space of the KC as the set 
    𝒟𝒮(a^n):= {  L^n-3 L_n-k-1∈[|L_n-k-a_n-k|,L_n-k+a_n-k] ∩[R_n-k-1^min,     R_n-k-1^max],  1≤ k ≤ n-3 }.




 A configuration β^n-1 contained in , that is it satisfies equation (<ref>), if and only if its diagonals are contained within 𝒟𝒮(a^n). Moreover, the the angle β_k is related to the angles β^k-1 by the formula 

    2 a_k L_k-1sin( β_k + Φ_k-1( β^k-1)  ) =  L_k^2  - a_k^2 - L_k-1^2,

for 1 ≤ k ≤ n-1, whenever Φ_k-1 is defined. If not, the relation between β_k and β^k-1 is given by (<ref>), where n-1 and n-2 are replaced by k and k-1. Note that for k=1 we set L_0 := 0 and Φ_0 :=0 so that β_1 is an arbitrary angle. 


 Assume β^n-1 is a circular configuration and thus solves equation (<ref>). We will manipulate equation (<ref>) to show that the vector of diagonal lengths L^n-2 is indeed an element of 𝒟𝒮(a^n). We start by showing  that the diagonal L_n-2 is contained  in [|L_n-1 - a_n-1|, L_n-1 - a_n-1| ] ∩ [     R_n-2^min ,     R_n-2^max]. For the remaining diagonals the analogous statement for L_n-k-1 then follows inductively by applying the same arguments.


Although it is very tempting to apply addition formula (<ref>) to this equation we have to  deal with special cases before we can do so.


X_n-2≠ 0 but Y_n-2 = 0. In this case, according to (<ref>) we obtain the equation  
    2 a_n-1cos(β_n-1 )  X_n-2 + L_n-2^2  =  L_n-1^2 - a_n-1^2.

solving for β_n-1 yields  

    cos(β_n-1 )     =   L_n-1^2 - L_n-2^2- a_n-1^2 / 2 a_n-1  X_n-2.

The Latter equation can be solved for β_n-1 if iff the square of its right side is lower or equal to one. Using Since in the considered case X_n-2^2 = L_n-2^2,  this leads to the inequality   

    (   L_n-1^2 - L_n-2^2- a_n-1^2 )^2 ≤  4 a_n-1^2 L_n–2^2,

hat the inequality is satisfied for L_n-2∈ [ |L_n-1 -a_n-1|,  L_n-1 +a_n-1 ]. Additionally, since L_n-2 is a diagonal length also L_n-2∈ [R_n-2^min,     R_n-2^max] has to be satisfied. The case when X_n-2 = 0 and Y_n-2≠ 0 is analogously.
       

The case L_n-2 = X_n-2 = Y_n-2 = 0 can only  occur for a_n = L_n-1 = a_n-1. Here β_n-1 is an arbitrary value, and R_n-2^min = 0 since L_n-2 = 0  is a diagonal of the circular configuration. Also 0 = L_n-2∈ [ |L_n-1 -a_n-1|,  L_n-1 +a_n-1 ] = [0, 2 a _n ] is satisfied.
 

If X_n-2≠ 0 and  X_n-2≠ 0 we can apply the addition theorem to formula (<ref>) and obtain   

    2 a_n-1√(  X_n-2^2 + Y_n-2^2 ) sin( β_n-1 + Φ(β^n-2)  ) + L_n-2^2  = L_n-1^2 - a_n-1^2,

which can be rewritten as 

    2 a_n-1 L_n-2sin( β_n-1 + Φ(β^n-2)  ) + L_n-2^2  = L_n-1^2 - a_n-1^2.

Repeating the arguments outlined above We conclude  that this equation can only have a solution if (<ref>) is satisfied and thus also in this case L_n-2∈  [ |L_n-1 -a_n-1|,  L_n-1 +a_n-1 ] and L_n-2∈ [R_n-2^min,     R_n-2^max]. Thus, repeating the arguments it follows that L^n-3∈𝒟𝒮(a^n).

       
Conversely, if L^n-3∈𝒟𝒮(a^n) then  equation (<ref>) can be solved for β_k if Φ_k-1 is defined. If not, equation (<ref>) can be used alternatively, with n-1, n-2 replaced by k and k-1. Repeating this procedure gives a circular configuration β^n-1. 


Theorem  <ref> outlines a procedure how the entries of elements L^n-3∈𝒟𝒮(a^n) can be determined recursively. This leads to a sampling strategy for the whole diagonal space. Moreover, equation (<ref>) explicitly describes how angles can be obtained from  the diagonals. We shortly summarize this in more detail in order to emphasize the progress of this work in comparison with the existing literature: 


  * The explicit sampling procedure for 𝒟𝒮(a^n) is an advance over the previous methods  <cit.>, where diagonals are obtained by linear programming. 

  * Looking more closely at the definition of 𝒟𝒮(a^n), one finds that the diagonal space is the intersection of a polytope and a cuboid.  More precisely, if 

    (a^n):={(L_2,…,L_n-2)∈ℝ^n-3_≥ 0 L_n-k-1∈[|L_n-k-a_n-k|,L_n-k+a_n-k]},

denotes the polytope defined by nested intervals, then

    𝒟𝒮(a^n) = (a^n) ∩(a^n),   with  (a^n) :=  ∏_1≤ k ≤ n-3[  R_n-k-1^min,   R_n-k-1^max].

This is a new representation of the diagonal space. 

Although the derived sampling procedure already provides a useful description of the diagonal space 𝒟𝒮(a^n),  we will investigate 𝒟𝒮(a^n) further in  section <ref>. 





§ FURTHER ANALYSIS OF CIRCULAR CONFIGURATIONS


In this section we will study the domain 𝒟𝒮(a^n) = (a^n) ∩(a^n), which naturally appears when we compute circular configurations, in more detail. After that we will have a closer look on  equation (<ref>), that has to be solved repeatedly to obtain the joint angles once diagonal length are computed.     




 §.§ The domain of Diagonals


The space 𝒟𝒮(a^n) =   (a^n) ∩(a^n) is the intersection of a polytope with a cuboid. We will see that the polytope can be parameterized if new variables    
U_2, … U_n-2 are introduced. This new insight is interesting from a mathematical point of view and could prove to be a potentially useful tool for further investigations of diagonal space. We will discuss it in more detail below.      


 Let (a^n) be given as in (<ref>). Define new variables U_2, …, U_n-2 according to the equation    

    L_n-k-1^2 =  U_n-k-1 + a_n-k^2   + L_n - k^2,     1 ≤ k ≤ n-3.

Then, for 1 ≤ k ≤ n-3, the U_k satisfy the following system of inequalities:

    | U_n-k-1 |  ≤ T_n-k(  U_n-k, …, U_n-2  ) := 2 a_n-k√(∑_j = n-k^n-2 U_j + ∑_j = n-k+1^n a_j^2 )

Note that for k=1  the value of the empty sum is zero and T_n-1:= 2a_n a_n-1.  




The inequalities defining (a^n) are equivalent to  

    (L_n-k-a_n-k)^2   ≤ L_n-k-1^2  ≤( L_n-k+a_n-k)^2,

 which is equivalent to 

    -2 L_n-k a_n-k≤ U_n-k-1≤ 2 L_n-k a_n-k

For any n it can be shown inductively that applying the substitution (<ref>) repeatedly equation to (<ref>) and thereby exploiting that L_n-1 = a_n yields system (<ref>). 



We define the transformed diagonal space.    

 The transformed polytope space is defined as the set 

    𝒫𝒰(a^n) = {   U^n-3 := (U_2, …, U_n-2) ∈^n-3 the entries of U^n-3 satisfy (<ref>)}



We will illustrate the transformation of the diagonal space for CKCs with five links.   


[CKC with five links] We consider the CKC with fife links a_1, …, a_5. In this case the inequalities defining 𝒫(a^5) are given by

    (a_5 - a_4)^2 ≤ L_3^2  ≤  (a_5 + a_4)^2,      (L_3 -a_3)^2 ≤ L_2^2  ≤ (L_3 +a_3)^2.

Expanding the squares and  using the substitutions L_2^2 = U_2 + a_3^2 + L_3^2 and L_3^2 = U_3 + a_4^2 + a_5^2 we obtain 

    -2a_5 a_4 ≤ U_3  ≤ 2a_5 a_4,       -2 L_3 a_3   ≤ U_2 ≤  2 L_3 a_3.

Using that L_3 = √(U_3 +a_4^2 + a_5^2) gives the inequalities 

    -2a_5 a_4 ≤ U_3  ≤ 2a_5 a_4,       -2  a_3 √(U_3 +a_4^2 + a_5^2)≤ U_2 ≤  2  a_3 √(U_3 +a_4^2 + a_5^2).

that define 𝒫𝒰(a^5). We observe that the transformed diagonal space can be described  by a parameter domain that is the cube [-1,1]^2.  More precisely,  𝒫𝒰(a^5) is the image of the map Γ[-1,1]^2 →𝒫𝒰(a^5) given by    
 
 [ s_2; s_3 ] ↦[                                2 a_4 a_5 s_2; 2 a_3 s_3 √( 2 a_4 a_5 s_1  + a_4^2 +a_5^2 ) ] = 
[ U_3(s_2, s_3); U_2(s_2, s_3) ]. 

Here it would be interesting to further investigate under which conditions Γ is actually a parameterization in the  differential geometric sense.  Parameters U_2, U_3 are related to the diagonals L_2, L_3 by the following equations 

L_3(s_2, s_3) =  √(   U_3(s_2, s_3) + a_4^2 + a_5^2)    and   L_2(s_2, s_3) =  √(   U_2(s_2, s_3) + a_3^2 + L_3^2) 

which are derived from (<ref>) for n = 5. In order for the diagonals obtained in this way to lie in 𝒟𝒮(a^5 ), the inequalities 

max{ a_1 -(a_2+ a_3), a_2 -(a_1+ a_3), a_3 -(a_1+ a_2), 0  } ≤    L_2(s_2,s_3) ≤a_1 +a_2 +a_3 

max{ a_1 -a_2, a_2 -a_1, 0  } ≤    L_3(s_2,s_3) ≤a_1 +a_2 

must still be fulfilled.        

The inequalities (<ref>) restrict the range of parameters s_2, s_3 to a sub domain of [-1,1]^2. Nevertheless, we think that further investigation of the mapping Γ, which can be defined analogously for a CKC with n links, is worthwhile for the following reasons: 
 

  * Numerical examples, see section <ref>, indicate that is possible that 𝒫𝒰(a^n) ⊂𝒬(a^n). In this cases no further restrictions are imposed and Γ can be used to obtain the diagonals.

  * A further investigation of the inequalities (<ref>) for the general case is interesting from a mathematical point of view and could also lead to simple parameter ranges of Γ, at least for special link lengths.          






 §.§ Flipping over lines through diagonals of a CKC
 

By theorem  <ref> a vector of joint angles β^n-1 can be computed from L^n-3∈𝒟𝒮(a^n ) by the formula   

    2 a_k L_k-1sin( β_k + Φ_k-1( β^k-1)  ) =  L_k^2  - a_k^2 - L_k-1^2 .

Clearly, solving for β_k is not unique. Each time we solve for β_k we have to decide which one of two preimages we chose. In order to denote which choice has been made we introduce the vector ^k = (_2, …, _k) ∈{0,1}^k-1 and refer to it as the orientation vector. Deending on the preimage that is taken when solving    (<ref>) the solutions for _k ∈{0,1} are given by     

β_k^_k  =  π_k + (-1)^_k  S_k - Φ_k-1(β^k-1, ^k-1 ).   

for 2 ≤ k ≤ n-1. Note, that in (<ref>) we used the following abbreviations. The term   

    S_k  = arcsin(   L_k^2  - a_k^2 - L_k-1^2  /  2a_k L_k-1)

and the vector  

    β^n, ^k = ( β_1, β_2^_2, …, β_n-k^_n-k),
 
to indicate the choice of preimages. Note that the first entry β_1  of the vector has no exponent, which will become clear after following geometric interpretation of formula (<ref>).
  


Summarizing, for each L^n-3∈𝒟𝒮(a^n) and a ∈{0,1}^n-2 we obtain a circular configuration 

    β^n-1,^n-1 = (β_1, β_2^_2, …, β_n-1^_n-1)
 
by formula (<ref>), if Φ_k-1 is defined. If not, we refer to (<ref>) as in theorem <ref>. There is a geometric interpretation for the the exponent _k in equation (<ref>). It describes how the link a_k is flipped over the line running through the origin and the end point of link a_k-1,  see Figure <ref>. 

Choosing the exponent _k corresponds to the choice of a triangle orientation in <cit.>. It is now clear that the angle β_1 has no exponent as β_1, since the link a_1 that is fixed to the origin encloses an arbitrary angle with the x-axis.   



§ NUMERICAL SIMULATIONS



In this section we provide numerical examples that demonstrate the validity of the methods developed in this work. For illustrative purposes we will consider CKCs with five and six links. For CKCs with five links we will depict the spaces 𝒟𝒮(a^n). We will randomly choose elements from the diagonal spaces and compute the corresponding circular configurations by Theorem <ref> and depict them in Figures <ref> and  <ref>.    



 §.§ CKCs with five and six links


First we will consider  CKCs with n=5 links. The domains 𝒟𝒮(a^5) are depicted for two CKCs in Figure <ref>. Figure <ref> shows 10 random circular configurations for these CKCs, where the length of the last link is equal to the radius of the depicted circle. The configurations have been obtained for the orientation vector ^4 = (_2, _3, _4) = (0,0,0).      


Finally, we give examples for CKSs with six links. Figure <ref> shows random configurations for CKCs with six links for the orientation vector  ^5 = (_2, _3,_4 ,_5 ) = (0, 0,0 ,0 ).






§ CONCLUSION AND FUTURE WORK



We have developed a new method for computing configurations in terms of joint angles of a CKC by a systematic procedure. Our approach does not require the solution of a system of linear inequalities by linear programming, nor does it rely on probabilistic methods. Numerical examples demonstrate the validity of the proposed work. We expect that the described method can be useful in tasks such as motion planning for CKCs. We anticipate that it will be an interesting approach for future work to further investigate the diagonal space 𝒟𝒮(a^n). Here it would be interesting to investigate how special designs for CKCs are reflected in the geometry of the diagonal space. Another interesting question is under what conditions 𝒫(a^n) ⊂𝒬(a^n) holds. Since it seems that in this situation 𝒫𝒰(a^n) and thus 𝒟𝒮(a^n) can be parameterized using Γ. A further investigation of Γ would then be another interesting line of research. 





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