# Crouch Gait Analysis and Visualization Based on Gait Forward and Inverse Kinematics
## 1 Introduction  
Human gait analysis (GA) is the systematic study of human walking [1]. Recently, instrumentation and computing technologies used to measure, process, and analyze body kinetics and kinematics have improved the efficiency of this assessment tool [2]. In clinical settings, gait analysis through kinetics and kinematics has been used to assess the effects of hip arthroplasty [3], stroke [4; 5], osteoarthritis [6], Achilles tendinopathy [7], rheumatoid arthritis (RA) [8], inversion sprains [9], and Parkinson's disease [10; 11], among others. Human motion estimation sensing is a challenging issue for gait analysis. For example, in [12], an inertial wearable is used to estimate model-based inverse kinematics. In the same sense, other studies have used Magnetic Resonance Imaging (MRI) [13] and motion capture marker-based systems [14; 15] to collect gait data in patients with cerebral palsy [13]. Crouch gait (CG) is one of the most common gait abnormalities, and is observed in people with cerebral palsy [16]; it is mainly characterized by excessive flexion of the hip, knee, and ankle [17; 18; 19]. In addition, crouch gait is distinguished in persons with hemiplegia after ischemic stroke [20]. CG causes joint pain [21] and arthritis [22], and is less efficient than a normal gait pattern [23]. Most of the works in the literature define a crouch gait only based on the knee position. Furthermore, kinematic parameters are well-defined when objective values of knee flexion are provided. However, few papers have considered these data and both lower limbs during the analysis [24], and consider the pathological gaitpatterns only in the sagittal plane. Other anatomical planes which could allow a crouch gait to be determined [18], such as the frontal and transverse planes, have not been considered. Gait kinematics is the study of the mechanical movements of the musculoskeletal system during walking without considering the forces and moments that originate it [1]. The two types of kinematic problems are classed as forward and inverse kinematics. Forward kinematics is the calculation of the end point of a linked structure from the known position of the joint parameters and segment lengths. On the other hand, inverse kinematics (IK) is the calculation of the joint parameters from the known position of the endpoint and segment lengths [25]. In robotics, inverse kinematics is used to determine the robot's capability, efficiency, and accuracy. For this purpose, analytical and numerical approaches have been developed to solve this complex problem. Analytical methods include closed-form and algebraic elimination-based algorithms [26], while numerical algorithms such as Newton-Raphson [27], Jacobian transpose [28], and damped least squares [29] converge to a single solution for IK. In [14], an inverse kinematics method for lower limb kinematics over the sagittal plane using Groebner Basis theory is proposed. For robotic manipulators, these methods fail with complex joint geometries or can produce run-time errors. In addition, quaternions have been used to perform orientations and rotations of objects in 3D space [30] in areas such as computer graphics, multirotor tracking, control approaches, and the kinematics and dynamics of rigid bodies [31]. The most widely used methods in robot kinematics are homogeneous transformation and Denavit-Hartenberg (DH). However, the free representation and gimbal lock avoidance of quaternions have led to their use being optimal for this purpose [32]. The conventional gait model (CGM) [33] is very useful in clinical gait laboratories. In direct kinematics (DK) methods, CGM is used to calculate joint kinematics. Meanwhile, for inverse kinematics (IK), musculoskeletal modeling approaches to obtain joint angles are used. It is recommended that the same anatomical model be used for kinematic and musculoskeletal analysis to ensure consistency between both spaces [13].