For robot manipulators subject to unmeasurable/uncertain plant parameters, this article designs a new adaptive motion controller, which ensures positioning errors to converge to zero and provides accurate gravity compensation. Meanwhile, specific motion constraints are also satisfied during the entire control process. Additionally, the proposed controller is further extended to address output feedback control without velocity measurement/numerical differential operations. A useful feature of this article is that neither complicated gain constraints nor the upper/lower bounds of model parameters/matrices in the dynamics are required in controller design and analysis, which greatly facilitates practical applications. Meanwhile, by introducing a nonlinear auxiliary term (related to motion constraints and error signals) into the proposed controllers, all links accurately reach their desired positions without exceeding the preset constraints, while the gravity vector is estimated online to eliminate static errors. Additionally, the asymptotic stability of the system equilibrium point is strictly proven; more importantly, the difficulty of stability analysis is significantly decreased based on the elaborately constructed Lyapunov function candidate. Compared with existing controllers, the main merits of the designed control schemes include fewer control gain conditions, more concise closed-loop stability analysis, and higher safety satisfying specific constraints. Finally, some hardware experiments are carried out to validate the performance of the presented controllers.