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A strategy combining an improved rapidly-exploring random tree (RRT) method with the artificial potential field (APF) method is proposed for the path planning of flexible tensegrity manipulators considering typical complex constraints, such as structural stability, obstacle avoidance, and cable no-slackening. The relationship between the node displacements and the elongations of active members is established for the kinematic analysis of tensegrity manipulators. The rest lengths of active members are taken as the design variables for the generation of random tree. The guide node is utilized for goal-biased samplings to expedite the exploration toward the final configuration, while the APF method is introduced to ensure that the obtained elongations of active members can satisfy the constraint of obstacle avoidance. The proportion of random and goal-biased samplings is adaptively adjusted based on the sampling success rate. Futile random samplings can be significantly reduced by dynamically modifying the random sampling range according to the obtained configuration nearest to the final configuration. The computational procedure of the proposed method is presented. An illustrative flexible tensegrity manipulator is employed to demonstrate the adaptability of the proposed path planning method to complex constraints, especially structural stability.