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This paper presents a novel continuum robot (OctRobot-I) that has controllable stiffness variation capability in both the transverse and axial directions. Robot design, stiffness variation analysis and experimental testing are discussed in detail. Stiffness models based on the Euler-Bernoulli beam theory are developed, and then four static deflection cases are analysed. Experiments are conducted with two types of layer jamming sheaths (overlap numbers <inline-formula> <tex-math notation="LaTeX">$n =3$ </tex-math></inline-formula>, 5) and four different vacuum pressures (0kPa, 25kPa, 50kPa, 75kPa) at three different bending angles (0&#x00B0;, 90&#x00B0;, 180&#x00B0;). The results demonstrate that the stiffness changing tendency is in compliance with the derived models and show that the robot has a good stiffness variable capability. With the jamming sheath of <inline-formula> <tex-math notation="LaTeX">$n =3$ </tex-math></inline-formula>, the stiffness ranges (ratios) are 36.4 to 241.7 N/m (6.6) and 92.9 to <inline-formula> <tex-math notation="LaTeX">$19.3\times 10^{3}$ </tex-math></inline-formula> N/m (207.8) in the transverse and axial directions, respectively. With the jamming sheath of <inline-formula> <tex-math notation="LaTeX">$n =5$ </tex-math></inline-formula>, the stiffness ranges (ratios) are 65.7 to 398.3 N/m (6.1) and 106.7 to <inline-formula> <tex-math notation="LaTeX">$20.8\times 10^{3}$ </tex-math></inline-formula> N/m (194.9) in the transverse and axial directions, respectively. Additionally, the actuating and gripping experiments demonstrate that this robot has good performance in real-world applications.