Generating Optimized Trajectories for Robotic Spray Painting

In the manufacturing industry, spray painting is often an important part of the manufacturing process. Especially in the automotive industry, the perceived quality of the final product is closely linked to the exactness and smoothness of the painting process. For complex products or low batch size production, manual spray painting is often used. But in large scale production with a high degree of automation, the painting is usually performed by industrial robots. There is a need to improve and simplify the generation of robot trajectories used in industrial paint booths. A novel method for spray paint optimization is presented, which can be used to smooth out a generated initial trajectory and minimize paint thickness deviations from a target thickness. The smoothed out trajectory is found by solving, using an interior point solver, a continuous non-linear optimization problem. A two-dimensional reference function of the applied paint thickness is selected by fitting a spline function to experimental data. This applicator footprint profile is then projected to the geometry and used as a paint deposition model. After generating an initial trajectory, the position and duration of each trajectory segment are used as optimization variables. The primary goal of the optimization is to obtain a paint applicator trajectory, which would closely match a target paint thickness when executed. The algorithm has been shown to produce satisfactory results on both a simple 2-dimensional test example, and a non-trivial industrial case of painting a tractor fender. The resulting trajectory is also proven feasible to be executed by an industrial robot

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean\u2013hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

A Novel Perception and Semantic Mapping Method for Robot Autonomy in Orchards

In this work, we propose a novel framework for achieving robotic autonomy in orchards. It consists of two key steps: perception and semantic mapping. In the perception step, we introduce a 3D detection method that accurately identifies objects directly on point cloud maps. In the semantic mapping step, we develop a mapping module that constructs a visibility graph map by incorporating object-level information and terrain analysis. By combining these two steps, our framework improves the autonomy of agricultural robots in orchard environments. The accurate detection of objects and the construction of a semantic map enable the robot to navigate autonomously, perform tasks such as fruit harvesting, and acquire actionable information for efficient agricultural production

A motion planner for nonholonomic mobile robots

This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R^2 x S^1 of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning

주행계 및 지도 작성을 위한 3차원 확률적 정규분포변환의 정합 방법

학위논문 (박사)-- 서울대학교 대학원 : 공과대학 전기·컴퓨터공학부, 2019. 2. 이범희.로봇은 거리센서를 이용하여 위치한 환경의 공간 정보를 점군(point set) 형태로 수집할 수 있는데, 이렇게 수집한 정보를 환경의 복원에 이용할 수 있다. 또한, 로봇은 점군과 모델을 정합하는 위치를 추정할 수 있다. 거리센서가 수집한 점군이 2차원에서 3차원으로 확장되고 해상도가 높아지면서 점의 개수가 크게 증가하면서, NDT (normal distributions transform)를 이용한 정합이 ICP (iterative closest point)의 대안으로 부상하였다. NDT는 점군을 분포로 변환하여 공간을 표현하는 압축된 공간 표현 방법이다. 분포의 개수가 점의 개수에 비해 월등히 작기 때문에 ICP에 비해 빠른 성능을 가졌다. 그러나 NDT 정합 기반 위치 추정의 성능을 좌우하는 셀의 크기, 셀의 중첩 정도, 셀의 방향, 분포의 스케일, 대응쌍의 비중 등 파라미터를 설정하기가 매우 어렵다. 본 학위 논문에서는 이러한 어려움에 대응하여 NDT 정합 기반 위치 추정의 정확도를 향상할 수 있는 방법을 제안하였다. 본 논문은 표현법과 정합법 2개 파트로 나눌 수 있다. 표현법에 있어 본 논문은 다음 3개 방법을 제안하였다. 첫째, 본 논문에서는 분포의 퇴화를 막기 위해 경험적으로 공분산 행렬의 고유값을 수정하여 공간적 형태의 왜곡을 가져오는 문제점과 고해상도의 NDT를 생성할 때 셀당 점의 개수가 감소하며 구조를 반영하는 분포가 형성되지 않는 문제점을 주목했다. 이를 해결하기 위하여 각 점에 대해 불확실성을 부여하고, 평균과 분산의 기대값으로 수정한 확률적 NDT (PNDT, probabilistic NDT) 표현법을 제안하였다. 공간 정보의 누락 없이 모든 점을 분포로 변환한 NDT를 통해 향상된 정확도를 보인 PNDT는 샘플링을 통한 가을을 가능하도록 하였다. 둘째, 본 논문에서는 정육면체를 셀로 다루며, 셀을 중심좌표와 변의 길이로 정의한다. 또한, 셀들로 이뤄진 격자를 각 셀의 중심점 사이의 간격과 셀의 크기로 정의한다. 이러한 정의를 토대로, 본 논문에서는 셀의 확대를 통하여 셀을 중첩시키는 방법과 셀의 간격 조절을 통하여 셀을 중첩시키는 방법을 제안하였다. 본 논문은 기존 2D NDT에서 사용한 셀의 삽입법을 주목하였다. 단순입방구조를 이루는 기존 방법 외에 면심입방구조와 체심입방구조의 셀로 이뤄진 격자가 생성하였다. 그 다음 해당 격자를 이용하여 NDT를 생성하는 방법을 제안하였다. 또한, 이렇게 생성된 NDT를 정합할 때 많은 시간을 소요하기 때문에 대응쌍 검색 영역을 정의하여 정합 속도를 향상하였다. 셋째, 저사양 로봇들은 점군 지도를 NDT 지도로 압축하여 보관하는 것이 효율적이다. 그러나 로봇 포즈가 갱신되거나, 다개체 로봇간 랑데뷰가 일어나 지도를 공유 및 결합하는 경우 NDT의 분포 형태가 왜곡되는 문제가 발생한다. 이러한 문제를 해결하기 위하여 NDT 재생성 방법을 제안하였다. 정합법에 있어 본 논문은 다음 4개 방법을 제안하였다. 첫째, 점군의 각 점에 대해 대응되는 색상 정보가 제공될 때 색상 hue를 이용한 향상된 NDT 정합으로 각 대응쌍에 대해 hue의 유사도를 비중으로 사용하는 목적함수를 제안하였다. 둘째, 본 논문은은 다양한 크기의 위치 변화량에 대응하기 위한 다중 레이어 NDT 정합 (ML-NDT, multi-layered NDT)의 한계를 극복하기 위하여 키레이어 NDT 정합 (KL-NDT, key-layered NDT)을 제안하였다. KL-NDT는 각 해상도의 셀에서 활성화된 점의 개수 변화량을 척도로 키레이어를 결정한다. 또한 키레이어에서 위치의 추정값이 수렴할 때까지 정합을 수행하는 방식을 취하여 다음 키레이어에 더 좋은 초기값을 제공한다. 셋째, 본 논문은 이산적인 셀로 인해 NDT간 정합 기법인 NDT-D2D (distribution-to-distribution NDT)의 목적 함수가 비선형이며 국소 최저치의 완화를 위한 방법으로 신규 NDT와 모델 NDT에 독립된 스케일을 정의하고 스케일을 변화하며 정합하는 동적 스케일 기반 NDT 정합 (DSF-NDT-D2D, dynamic scaling factor-based NDT-D2D)을 제안하였다. 마지막으로, 본 논문은 소스 NDT와 지도간 증대적 정합을 이용한 주행계 추정 및 지도 작성 방법을 제안하였다. 이 방법은 로봇의 현재 포즈에 대한 초기값을 소스 점군에 적용한 뒤 NDT로 변환하여 지도 상 NDT와 가능한 한 유사한 NDT를 작성한다. 그 다음 로봇 포즈 및 소스 NDT의 GC (Gaussian component)를 고려하여 부분지도를 추출한다. 이렇게 추출한 부분지도와 소스 NDT는 다중 레이어 NDT 정합을 수행하여 정확한 주행계를 추정하고, 추정 포즈로 소스 점군을 회전 및 이동 후 기존 지도를 갱신한다. 이러한 과정을 통해 이 방법은 현재 최고 성능을 가진 LOAM (lidar odometry and mapping)에 비하여 더 높은 정확도와 더 빠른 처리속도를 보였다.The robot is a self-operating device using its intelligence, and autonomous navigation is a critical form of intelligence for a robot. This dissertation focuses on localization and mapping using a 3D range sensor for autonomous navigation. The robot can collect spatial information from the environment using a range sensor. This information can be used to reconstruct the environment. Additionally, the robot can estimate pose variations by registering the source point set with the model. Given that the point set collected by the sensor is expanded in three dimensions and becomes dense, registration using the normal distribution transform (NDT) has emerged as an alternative to the most commonly used iterative closest point (ICP) method. NDT is a compact representation which describes using a set of GCs (GC) converted from a point set. Because the number of GCs is much smaller than the number of points, with regard to the computation time, NDT outperforms ICP. However, the NDT has issues to be resolved, such as the discretization of the point set and the objective function. This dissertation is divided into two parts: representation and registration. For the representation part, first we present the probabilistic NDT (PNDT) to deal with the destruction and degeneration problems caused by the small cell size and the sparse point set. PNDT assigns an uncertainty to each point sample to convert a point set with fewer than four points into a distribution. As a result, PNDT allows for more precise registration using small cells. Second, we present lattice adjustment and cell insertion methods to overlap cells to overcome the discreteness problem of the NDT. In the lattice adjustment method, a lattice is expressed as the distance between the cells and the side length of each cell. In the cell insertion method, simple, face-centered-cubic, and body-centered-cubic lattices are compared. Third, we present a means of regenerating the NDT for the target lattice. A single robot updates its poses using simultaneous localization and mapping (SLAM) and fuses the NDT at each pose to update its NDT map. Moreover, multiple robots share NDT maps built with inconsistent lattices and fuse the maps. Because the simple fusion of the NDT maps can change the centers, shapes, and normal vectors of GCs, the regeneration method subdivides the NDT into truncated GCs using the target lattice and regenerates the NDT. For the registration part, first we present a hue-assisted NDT registration if the robot acquires color information corresponding to each point sample from a vision sensor. Each GC of the NDT has a distribution of the hue and uses the similarity of the hue distributions as the weight in the objective function. Second, we present a key-layered NDT registration (KL-NDT) method. The multi-layered NDT registration (ML-NDT) registers points to the NDT in multiple resolutions of lattices. However, the initial cell size and the number of layers are difficult to determine. KL-NDT determines the key layers in which the registration is performed based on the change of the number of activated points. Third, we present a method involving dynamic scaling factors of the covariance. This method scales the source NDT at zero initially to avoid a negative correlation between the likelihood and rotational alignment. It also scales the target NDT from the maximum scale to the minimum scale. Finally, we present a method of incremental registration of PNDTs which outperforms the state-of-the-art lidar odometry and mapping method.1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Point Set Registration . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Incremental Registration for Odometry Estimation . . . . . . 16 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Preliminaries 21 2.1 NDT Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 NDT Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 NDT Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Transformation Matrix and The Parameter Vector . . . . . . . . . . . 27 2.5 Cubic Cell and Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Evaluation of Registration . . . . . . . . . . . . . . . . . . . . . . . 31 2.9 Benchmark Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Probabilistic NDT Representation 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Uncertainty of Point Based on Sensor Model . . . . . . . . . . . . . . 36 3.3 Probabilistic NDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Generalization of NDT Registration Based on PNDT . . . . . . . . . 40 3.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5.2 Evaluation of Representation . . . . . . . . . . . . . . . . . . 41 3.5.3 Evaluation of Registration . . . . . . . . . . . . . . . . . . . 46 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Interpolation for NDT Using Overlapped Regular Cells 51 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Lattice Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Crystalline NDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.1 Lattice Adjustment . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.2 Performance of Crystalline NDT . . . . . . . . . . . . . . . . 60 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5 Regeneration of Normal Distributions Transform 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 67 5.2.1 Trivariate Normal Distribution . . . . . . . . . . . . . . . . . 67 5.2.2 Truncated Trivariate Normal Distribution . . . . . . . . . . . 67 5.3 Regeneration of NDT . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.1 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.2 Subdivision of Gaussian Components . . . . . . . . . . . . . 70 5.3.3 Fusion of Gaussian Components . . . . . . . . . . . . . . . . 72 5.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4.1 Evaluation Metrics for Representation . . . . . . . . . . . . . 73 5.4.2 Representation Performance of the Regenerated NDT . . . . . 75 5.4.3 Computation Performance of the Regeneration . . . . . . . . 82 5.4.4 Application of Map Fusion . . . . . . . . . . . . . . . . . . . 83 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6 Hue-Assisted Registration 91 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Preliminary of the HSV Model . . . . . . . . . . . . . . . . . . . . . 92 6.3 Colored Octree for Subdivision . . . . . . . . . . . . . . . . . . . . . 94 6.4 HA-NDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.5.1 Evaluation of HA-NDT against nhue . . . . . . . . . . . . . . 97 6.5.2 Evaluation of NDT and HA-NDT . . . . . . . . . . . . . . . 98 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Key-Layered NDT Registration 103 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 Key-layered NDT-P2D . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3.1 Evaluation of KL-NDT-P2D and ML-NDT-P2D . . . . . . . . 108 7.3.2 Evaluation of KL-NDT-D2D and ML-NDT-D2D . . . . . . . 111 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8 Scaled NDT and The Multi-scale Registration 113 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.2 Scaled NDT representation and L2 distance . . . . . . . . . . . . . . 114 8.3 NDT-D2D with dynamic scaling factors of covariances . . . . . . . . 116 8.4 Range of scaling factors . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.5.1 Evaluation of the presented method without initial guess . . . 122 8.5.2 Application of odometry estimation . . . . . . . . . . . . . . 125 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 9 Scan-to-map Registration 129 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 9.2 Multi-layered PNDT . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.3 NDT Incremental Registration . . . . . . . . . . . . . . . . . . . . . 132 9.3.1 Initialization of PNDT-Map . . . . . . . . . . . . . . . . . . 133 9.3.2 Generation of Source ML-PNDT . . . . . . . . . . . . . . . . 134 9.3.3 Reconstruction of The Target ML-PNDT . . . . . . . . . . . 134 9.3.4 Pose Estimation Based on Multi-layered Registration . . . . . 135 9.3.5 Update of PNDT-Map . . . . . . . . . . . . . . . . . . . . . 136 9.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10 Conclusions 142 Bibliography 145 초록 159 감사의 글 162Docto

Detail Enhancing Denoising of Digitized 3D Models from a Mobile Scanning System

The acquisition process of digitizing a large-scale environment produces an enormous amount of raw geometry data. This data is corrupted by system noise, which leads to 3D surfaces that are not smooth and details that are distorted. Any scanning system has noise associate with the scanning hardware, both digital quantization errors and measurement inaccuracies, but a mobile scanning system has additional system noise introduced by the pose estimation of the hardware during data acquisition. The combined system noise generates data that is not handled well by existing noise reduction and smoothing techniques. This research is focused on enhancing the 3D models acquired by mobile scanning systems used to digitize large-scale environments. These digitization systems combine a variety of sensors – including laser range scanners, video cameras, and pose estimation hardware – on a mobile platform for the quick acquisition of 3D models of real world environments. The data acquired by such systems are extremely noisy, often with significant details being on the same order of magnitude as the system noise. By utilizing a unique 3D signal analysis tool, a denoising algorithm was developed that identifies regions of detail and enhances their geometry, while removing the effects of noise on the overall model. The developed algorithm can be useful for a variety of digitized 3D models, not just those involving mobile scanning systems. The challenges faced in this study were the automatic processing needs of the enhancement algorithm, and the need to fill a hole in the area of 3D model analysis in order to reduce the effect of system noise on the 3D models. In this context, our main contributions are the automation and integration of a data enhancement method not well known to the computer vision community, and the development of a novel 3D signal decomposition and analysis tool. The new technologies featured in this document are intuitive extensions of existing methods to new dimensionality and applications. The totality of the research has been applied towards detail enhancing denoising of scanned data from a mobile range scanning system, and results from both synthetic and real models are presented

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean–hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Computing fast search heuristics for physics-based mobile robot motion planning

Mobile robots are increasingly being employed to assist responders in search and rescue missions. Robots have to navigate in dangerous areas such as collapsed buildings and hazardous sites, which can be inaccessible to humans. Tele-operating the robots can be stressing for the human operators, which are also overloaded with mission tasks and coordination overhead, so it is important to provide the robot with some degree of autonomy, to lighten up the task for the human operator and also to ensure robot safety. Moving robots around requires reasoning, including interpretation of the environment, spatial reasoning, planning of actions (motion), and execution. This is particularly challenging when the environment is unstructured, and the terrain is \textit{harsh}, i.e. not flat and cluttered with obstacles. Approaches reducing the problem to a 2D path planning problem fall short, and many of those who reason about the problem in 3D don't do it in a complete and exhaustive manner. The approach proposed in this thesis is to use rigid body simulation to obtain a more truthful model of the reality, i.e. of the interaction between the robot and the environment. Such a simulation obeys the laws of physics, takes into account the geometry of the environment, the geometry of the robot, and any dynamic constraints that may be in place. The physics-based motion planning approach by itself is also highly intractable due to the computational load required to perform state propagation combined with the exponential blowup of planning; additionally, there are more technical limitations that disallow us to use things such as state sampling or state steering, which are known to be effective in solving the problem in simpler domains. The proposed solution to this problem is to compute heuristics that can bias the search towards the goal, so as to quickly converge towards the solution. With such a model, the search space is a rich space, which can only contain states which are physically reachable by the robot, and also tells us enough information about the safety of the robot itself. The overall result is that by using this framework the robot engineer has a simpler job of encoding the \textit{domain knowledge} which now consists only of providing the robot geometric model plus any constraints