Deep learning (DL) has achieved great success in many applications, but it has been less well analyzed from the theoretical perspective. The unexplainable success of black-box DL models has raised questions among scientists and promoted the emergence of the field of explainable artificial intelligence (XAI). In robotics, it is particularly important to deploy DL algorithms in a predictable and stable manner as robots are active agents that need to interact safely with the physical world. This paper presents an analytic deep learning framework for fully connected neural networks, which can be applied for both regression problems and classification problems. Examples for regression and classification problems include online robot control and robot vision. We present two layer-wise learning algorithms such that the convergence of the learning systems can be analyzed. Firstly, an inverse layer-wise learning algorithm for multilayer networks with convergence analysis for each layer is presented to understand the problems of layer-wise deep learning. Secondly, a forward progressive learning algorithm where the deep networks are built progressively by using single hidden layer networks is developed to achieve better accuracy. It is shown that the progressive learning method can be used for fine-tuning of weights from convergence point of view. The effectiveness of the proposed framework is illustrated based on classical benchmark recognition tasks using the MNIST and CIFAR-10 datasets and the results show a good balance between performance and explainability. The proposed method is subsequently applied for online learning of robot kinematics and experimental results on kinematic control of UR5e robot with unknown model are presented.