Bir manipülatörün kinematik yeterliliği, o manipülatörün uç noktasının erişebildiği toplam küme olan çalışma uzayında keyfi seçilen her yönde öteleme ve yine keyfi seçilen her yönde dönme hareketi yapabilmesidir. Diğer bir değişle kinematik yetersiz manipulatörler, çalışma uzaylarında geçerli bütün konfigürasyonlara ulaşmak için gereken serbestlik derecelerinden (SD) daha azına sahip olan manipülatörlerdir. Üç boyutlu çalışma uzayı için bu durum, bir manipülatörün serbestlik derecesinin altıdan daha küçük olmasına karşı düşer. Çünkü üç boyutlu çalışma uzayına sahip kinematik yeterliliği olan bir manipülatörün uç noktası, üç boyutta dönme ve üç boyutta öteleme olmak üzere toplam altı boyutlu bir manifold tanımlar. Birlikte çalışan manipülatörlerden oluşan bir sistem üzerindeki kuvvet ve moment dağılımlarını hesaplayabilmek için sistemin Jakobiyen matrisinin sütunlarının bütün kombinasyonları bu manifoldu tamamıyla tarayabilmelidir. Bundan dolayı literatürde genellikle manipülatörlerin kinematik yeterliliği ve tekil durumda olmamaları bu problemin çözümüne ön koşul olarak getirilmektedir. Birlikte çalışan manipülatörlerin dinamik analizinde kinematik yeterlilik ön koşulunun kaldırılması amacıyla bu çalışmada, manipülatörlerin taşıdığı yükü bir mobil platform olarak modelleyerek sisteme altı serbestlik derecesi kazandırmak ve bunun yeterli olmadığı durumlarda sisteme “sözde eklem” eklemek olarak özetlenebilecek bir yöntem tanıtılmaktadır. Bu yöntemin holonomik olmayan sistemlere uygulaması da araç dinamiğinde tekerlek ve yol arasındaki kuvvetlerin hesaplanabilmesi anlamına geldiği için otomotiv endüstrisinde ayrıca önem taşır.

Precision and load capacity are among the reasons why more than one manipulator may be used to perform a common task. In such cooperation, propagating force and torque within and among manipulators cannot be computed unless the columns of the System- Jacobian matrix span the operational space completely. This can be made sure as long as the end-effector of each manipulator in the system independently move and rotate at any direction in the operational space. In order to have a threedimensional task space for rotation and translation, a manipulator needs to have at least six Degrees of Freedom (DOF). Then the Jacobian of such manipulator is required to have columns that span a sixdimensional manifold provided that it is not at a singular configuration. Many industrial applications do not require the full kinematic capability to move and rotate the tip point of the manipulator at any direction. In regards to cost, manufacturing, and compactness, any DOF not necessary for the task should be avoided unless redundancy is needed for both operational space and joint space controls such as obstacle or joint limit avoidance problems. With this motivation, kinematic deficiency is defined as the lack of ability of the endeffector to independently move towards and rotate around any vector in the operational space. Therefore, kinematically deficient manipulators are those that have fewer degrees of freedom than necessary to achieve any admissible configuration in their operational space. This paper addresses the challenges associated with the computation of constrained forces at a cooperating manipulator system in the presence of kinematic deficiency. One may suggest eliminating the deficiency by reducing the size of the task space by removing the directions towards which the endeffector of the manipulator cannot move from the Jacobian. Considering that the range space of the Jacobian represents the space in which the endeffector of the manipulator is free to move. The space in which the tip of the manipulator cannot move is the one perpendicular to the range space. Therefore, it is the null space of the transpose of the Jacobian. The drawback of this method is that the computation of null space requires Singular Value Decomposition (SVD), which introduces instability due to the fact that singular vectors are not unique, and may introduce discontinuity. This drawback alone makes this methodology impractical, not to mention the cost associated with the numerical computation of SVD. These are the disadvantages of the numerical methods in general addressing this issue. Consequently, there is a need for an analytical approach on which our work particularly concentrates. We suggest two methods; “common load as mobile platform” and the “pseudo joint.” Common-loadas- mobile-platform method is essentially a modeling strategy. The basic idea behind it is to model the load carried by the manipulators as if it were the mobile platform carrying the manipulators. Hence, we need to re-label the body frames to reflect the new assignments of the end-effector and the base. The reason of this is to gain 6 DOF from the platform. In many cases, this additional 6 DOF together with appropriate constraint equations yield the constraint forces. For the other cases we propose the pseudo-joint method, which may be used in conjunction with the common-load-as-mobile-platform method. The pseudo-joint method is used at a kinematically deficient system where additional 6 DOF introduced by the “common load as mobile platform” method is not sufficient. In this case, we first augment the system with additional joints called pseudo-joints and then remove them from the system by computed constraints, which actually correspond to stress along the pseudo-joints. Therefore, for the price of stress computation, we obtain full propagation of forces and torques including the constrained ones throughout the system. These techniques can be applied to nonholonomic systems where the constraints are not integrable. We demonstrate the use of the algorithm on wheeled systems and show the constrained forces. In many applications it is critical to know the tireroad interaction forces, which determine how a vehicle turns, brakes and accelerates. Future work of this study includes multitrailer vehicle dynamics and control.