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A novel reconfigurable intelligent surface-aided multi-robot network is proposed, where multiple mobile robots are served by an access point (AP) through non-orthogonal multiple access (NOMA). The goal is to maximize the sum-rate of whole trajectories for the multi-robot system by jointly optimizing trajectories and NOMA decoding orders of robots, phase-shift coefficients of the RIS, and the power allocation of the AP, subject to predicted initial and final positions of robots and the quality of service (QoS) of each robot. To tackle this problem, an integrated machine learning (ML) scheme is proposed, which combines long short-term memory (LSTM)-autoregressive integrated moving average (ARIMA) model and dueling double deep Q-network (D$^{3}$QN) algorithm. For initial and final position prediction for robots, the LSTM-ARIMA is able to overcome the problem of gradient vanishment of non-stationary and non-linear sequences of data. For jointly determining the phase shift matrix and robots' trajectories, D$^{3}$QN is invoked for solving the problem of action value overestimation. Based on the proposed scheme, each robot holds an optimal trajectory based on the maximum sum-rate of a whole trajectory, which reveals that robots pursue long-term benefits for whole trajectory design. Numerical results demonstrated that: 1) LSTM-ARIMA model provides high accuracy predicting model; 2) The proposed D$^{3}$QN algorithm can achieve fast average convergence; and 3) RIS-NOMA networks have superior network performance compared to RIS-aided orthogonal counterparts.